diff --git a/doc/source/whatsnew/v0.24.1.rst b/doc/source/whatsnew/v0.24.1.rst
index 047404e93914b..8f2584934d13a 100644
--- a/doc/source/whatsnew/v0.24.1.rst
+++ b/doc/source/whatsnew/v0.24.1.rst
@@ -15,10 +15,40 @@ Whats New in 0.24.1 (February XX, 2019)
 These are the changes in pandas 0.24.1. See :ref:`release` for a full changelog
 including other versions of pandas.
 
+.. _whatsnew_0241.api:
+
+API Changes
+~~~~~~~~~~~
+
+Changing the ``sort`` parameter for :class:`Index` set operations
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+The default ``sort`` value for :meth:`Index.union` has changed from ``True`` to ``None`` (:issue:`24959`).
+The default *behavior*, however, remains the same: the result is sorted, unless
+
+1. ``self`` and ``other`` are identical
+2. ``self`` or ``other`` is empty
+3. ``self`` or ``other`` contain values that can not be compared (a ``RuntimeWarning`` is raised).
+
+This change will allow ``sort=True`` to mean "always sort" in a future release.
+
+The same change applies to :meth:`Index.difference` and :meth:`Index.symmetric_difference`, which
+would not sort the result when the values could not be compared.
+
+The `sort` option for :meth:`Index.intersection` has changed in three ways.
+
+1. The default has changed from ``True`` to ``False``, to restore the
+   pandas 0.23.4 and earlier behavior of not sorting by default.
+2. The behavior of ``sort=True`` can now be obtained with ``sort=None``.
+   This will sort the result only if the values in ``self`` and ``other``
+   are not identical.
+3. The value ``sort=True`` is no longer allowed. A future version of pandas
+   will properly support ``sort=True`` meaning "always sort".
+
 .. _whatsnew_0241.regressions:
 
 Fixed Regressions
-^^^^^^^^^^^^^^^^^
+~~~~~~~~~~~~~~~~~
 
 - Bug in :meth:`DataFrame.itertuples` with ``records`` orient raising an ``AttributeError`` when the ``DataFrame`` contained more than 255 columns (:issue:`24939`)
 - Bug in :meth:`DataFrame.itertuples` orient converting integer column names to strings prepended with an underscore (:issue:`24940`)
@@ -28,7 +58,7 @@ Fixed Regressions
 .. _whatsnew_0241.enhancements:
 
 Enhancements
-^^^^^^^^^^^^
+~~~~~~~~~~~~
 
 
 .. _whatsnew_0241.bug_fixes:
diff --git a/pandas/_libs/lib.pyx b/pandas/_libs/lib.pyx
index 4a3440e14ba14..f8875d60049b1 100644
--- a/pandas/_libs/lib.pyx
+++ b/pandas/_libs/lib.pyx
@@ -233,10 +233,11 @@ def fast_unique_multiple(list arrays, sort: bool=True):
             if val not in table:
                 table[val] = stub
                 uniques.append(val)
-    if sort:
+    if sort is None:
         try:
             uniques.sort()
         except Exception:
+            # TODO: RuntimeWarning?
             pass
 
     return uniques
diff --git a/pandas/core/indexes/api.py b/pandas/core/indexes/api.py
index 684a19c56c92f..6299fc482d0df 100644
--- a/pandas/core/indexes/api.py
+++ b/pandas/core/indexes/api.py
@@ -112,7 +112,7 @@ def _get_combined_index(indexes, intersect=False, sort=False):
     elif intersect:
         index = indexes[0]
         for other in indexes[1:]:
-            index = index.intersection(other, sort=sort)
+            index = index.intersection(other)
     else:
         index = _union_indexes(indexes, sort=sort)
         index = ensure_index(index)
diff --git a/pandas/core/indexes/base.py b/pandas/core/indexes/base.py
index 3d176012df22b..2fa034670e885 100644
--- a/pandas/core/indexes/base.py
+++ b/pandas/core/indexes/base.py
@@ -2245,18 +2245,37 @@ def _get_reconciled_name_object(self, other):
             return self._shallow_copy(name=name)
         return self
 
-    def union(self, other, sort=True):
+    def _validate_sort_keyword(self, sort):
+        if sort not in [None, False]:
+            raise ValueError("The 'sort' keyword only takes the values of "
+                             "None or False; {0} was passed.".format(sort))
+
+    def union(self, other, sort=None):
         """
         Form the union of two Index objects.
 
         Parameters
         ----------
         other : Index or array-like
-        sort : bool, default True
-            Sort the resulting index if possible
+        sort : bool or None, default None
+            Whether to sort the resulting Index.
+
+            * None : Sort the result, except when
+
+              1. `self` and `other` are equal.
+              2. `self` or `other` has length 0.
+              3. Some values in `self` or `other` cannot be compared.
+                 A RuntimeWarning is issued in this case.
+
+            * False : do not sort the result.
 
             .. versionadded:: 0.24.0
 
+            .. versionchanged:: 0.24.1
+
+               Changed the default value from ``True`` to ``None``
+               (without change in behaviour).
+
         Returns
         -------
         union : Index
@@ -2269,6 +2288,7 @@ def union(self, other, sort=True):
         >>> idx1.union(idx2)
         Int64Index([1, 2, 3, 4, 5, 6], dtype='int64')
         """
+        self._validate_sort_keyword(sort)
         self._assert_can_do_setop(other)
         other = ensure_index(other)
 
@@ -2319,7 +2339,7 @@ def union(self, other, sort=True):
             else:
                 result = lvals
 
-            if sort:
+            if sort is None:
                 try:
                     result = sorting.safe_sort(result)
                 except TypeError as e:
@@ -2342,14 +2362,19 @@ def intersection(self, other, sort=False):
         Parameters
         ----------
         other : Index or array-like
-        sort : bool, default False
-            Sort the resulting index if possible
+        sort : False or None, default False
+            Whether to sort the resulting index.
+
+            * False : do not sort the result.
+            * None : sort the result, except when `self` and `other` are equal
+              or when the values cannot be compared.
 
             .. versionadded:: 0.24.0
 
             .. versionchanged:: 0.24.1
 
-               Changed the default from ``True`` to ``False``.
+               Changed the default from ``True`` to ``False``, to match
+               the behaviour of 0.23.4 and earlier.
 
         Returns
         -------
@@ -2363,6 +2388,7 @@ def intersection(self, other, sort=False):
         >>> idx1.intersection(idx2)
         Int64Index([3, 4], dtype='int64')
         """
+        self._validate_sort_keyword(sort)
         self._assert_can_do_setop(other)
         other = ensure_index(other)
 
@@ -2402,7 +2428,7 @@ def intersection(self, other, sort=False):
 
         taken = other.take(indexer)
 
-        if sort:
+        if sort is None:
             taken = sorting.safe_sort(taken.values)
             if self.name != other.name:
                 name = None
@@ -2415,7 +2441,7 @@ def intersection(self, other, sort=False):
 
         return taken
 
-    def difference(self, other, sort=True):
+    def difference(self, other, sort=None):
         """
         Return a new Index with elements from the index that are not in
         `other`.
@@ -2425,11 +2451,22 @@ def difference(self, other, sort=True):
         Parameters
         ----------
         other : Index or array-like
-        sort : bool, default True
-            Sort the resulting index if possible
+        sort : False or None, default None
+            Whether to sort the resulting index. By default, the
+            values are attempted to be sorted, but any TypeError from
+            incomparable elements is caught by pandas.
+
+            * None : Attempt to sort the result, but catch any TypeErrors
+              from comparing incomparable elements.
+            * False : Do not sort the result.
 
             .. versionadded:: 0.24.0
 
+            .. versionchanged:: 0.24.1
+
+               Changed the default value from ``True`` to ``None``
+               (without change in behaviour).
+
         Returns
         -------
         difference : Index
@@ -2444,6 +2481,7 @@ def difference(self, other, sort=True):
         >>> idx1.difference(idx2, sort=False)
         Int64Index([2, 1], dtype='int64')
         """
+        self._validate_sort_keyword(sort)
         self._assert_can_do_setop(other)
 
         if self.equals(other):
@@ -2460,7 +2498,7 @@ def difference(self, other, sort=True):
         label_diff = np.setdiff1d(np.arange(this.size), indexer,
                                   assume_unique=True)
         the_diff = this.values.take(label_diff)
-        if sort:
+        if sort is None:
             try:
                 the_diff = sorting.safe_sort(the_diff)
             except TypeError:
@@ -2468,7 +2506,7 @@ def difference(self, other, sort=True):
 
         return this._shallow_copy(the_diff, name=result_name, freq=None)
 
-    def symmetric_difference(self, other, result_name=None, sort=True):
+    def symmetric_difference(self, other, result_name=None, sort=None):
         """
         Compute the symmetric difference of two Index objects.
 
@@ -2476,11 +2514,22 @@ def symmetric_difference(self, other, result_name=None, sort=True):
         ----------
         other : Index or array-like
         result_name : str
-        sort : bool, default True
-            Sort the resulting index if possible
+        sort : False or None, default None
+            Whether to sort the resulting index. By default, the
+            values are attempted to be sorted, but any TypeError from
+            incomparable elements is caught by pandas.
+
+            * None : Attempt to sort the result, but catch any TypeErrors
+              from comparing incomparable elements.
+            * False : Do not sort the result.
 
             .. versionadded:: 0.24.0
 
+            .. versionchanged:: 0.24.1
+
+               Changed the default value from ``True`` to ``None``
+               (without change in behaviour).
+
         Returns
         -------
         symmetric_difference : Index
@@ -2504,6 +2553,7 @@ def symmetric_difference(self, other, result_name=None, sort=True):
         >>> idx1 ^ idx2
         Int64Index([1, 5], dtype='int64')
         """
+        self._validate_sort_keyword(sort)
         self._assert_can_do_setop(other)
         other, result_name_update = self._convert_can_do_setop(other)
         if result_name is None:
@@ -2524,7 +2574,7 @@ def symmetric_difference(self, other, result_name=None, sort=True):
         right_diff = other.values.take(right_indexer)
 
         the_diff = _concat._concat_compat([left_diff, right_diff])
-        if sort:
+        if sort is None:
             try:
                 the_diff = sorting.safe_sort(the_diff)
             except TypeError:
diff --git a/pandas/core/indexes/datetimes.py b/pandas/core/indexes/datetimes.py
index ef941ab87ba12..9c46860eb49d6 100644
--- a/pandas/core/indexes/datetimes.py
+++ b/pandas/core/indexes/datetimes.py
@@ -602,19 +602,21 @@ def intersection(self, other, sort=False):
         Parameters
         ----------
         other : DatetimeIndex or array-like
-        sort : bool, default True
+        sort : False or None, default False
             Sort the resulting index if possible.
 
             .. versionadded:: 0.24.0
 
             .. versionchanged:: 0.24.1
 
-               Changed the default from ``True`` to ``False``.
+               Changed the default to ``False`` to match the behaviour
+               from before 0.24.0.
 
         Returns
         -------
         y : Index or DatetimeIndex
         """
+        self._validate_sort_keyword(sort)
         self._assert_can_do_setop(other)
 
         if self.equals(other):
diff --git a/pandas/core/indexes/interval.py b/pandas/core/indexes/interval.py
index 736de94991181..2c63fe33c57fe 100644
--- a/pandas/core/indexes/interval.py
+++ b/pandas/core/indexes/interval.py
@@ -1093,7 +1093,7 @@ def equals(self, other):
     def overlaps(self, other):
         return self._data.overlaps(other)
 
-    def _setop(op_name, sort=True):
+    def _setop(op_name, sort=None):
         def func(self, other, sort=sort):
             other = self._as_like_interval_index(other)
 
diff --git a/pandas/core/indexes/multi.py b/pandas/core/indexes/multi.py
index 16af3fe8eef26..14975dbbefa63 100644
--- a/pandas/core/indexes/multi.py
+++ b/pandas/core/indexes/multi.py
@@ -2879,30 +2879,47 @@ def equal_levels(self, other):
                 return False
         return True
 
-    def union(self, other, sort=True):
+    def union(self, other, sort=None):
         """
         Form the union of two MultiIndex objects
 
         Parameters
         ----------
         other : MultiIndex or array / Index of tuples
-        sort : bool, default True
-            Sort the resulting MultiIndex if possible
+        sort : False or None, default None
+            Whether to sort the resulting Index.
+
+            * None : Sort the result, except when
+
+              1. `self` and `other` are equal.
+              2. `self` has length 0.
+              3. Some values in `self` or `other` cannot be compared.
+                 A RuntimeWarning is issued in this case.
+
+            * False : do not sort the result.
 
             .. versionadded:: 0.24.0
 
+            .. versionchanged:: 0.24.1
+
+               Changed the default value from ``True`` to ``None``
+               (without change in behaviour).
+
         Returns
         -------
         Index
 
         >>> index.union(index2)
         """
+        self._validate_sort_keyword(sort)
         self._assert_can_do_setop(other)
         other, result_names = self._convert_can_do_setop(other)
 
         if len(other) == 0 or self.equals(other):
             return self
 
+        # TODO: Index.union returns other when `len(self)` is 0.
+
         uniq_tuples = lib.fast_unique_multiple([self._ndarray_values,
                                                 other._ndarray_values],
                                                sort=sort)
@@ -2917,19 +2934,21 @@ def intersection(self, other, sort=False):
         Parameters
         ----------
         other : MultiIndex or array / Index of tuples
-        sort : bool, default True
+        sort : False or None, default False
             Sort the resulting MultiIndex if possible
 
             .. versionadded:: 0.24.0
 
             .. versionchanged:: 0.24.1
 
-               Changed the default from ``True`` to ``False``.
+               Changed the default from ``True`` to ``False``, to match
+               behaviour from before 0.24.0
 
         Returns
         -------
         Index
         """
+        self._validate_sort_keyword(sort)
         self._assert_can_do_setop(other)
         other, result_names = self._convert_can_do_setop(other)
 
@@ -2940,7 +2959,7 @@ def intersection(self, other, sort=False):
         other_tuples = other._ndarray_values
         uniq_tuples = set(self_tuples) & set(other_tuples)
 
-        if sort:
+        if sort is None:
             uniq_tuples = sorted(uniq_tuples)
 
         if len(uniq_tuples) == 0:
@@ -2951,22 +2970,28 @@ def intersection(self, other, sort=False):
             return MultiIndex.from_arrays(lzip(*uniq_tuples), sortorder=0,
                                           names=result_names)
 
-    def difference(self, other, sort=True):
+    def difference(self, other, sort=None):
         """
         Compute set difference of two MultiIndex objects
 
         Parameters
         ----------
         other : MultiIndex
-        sort : bool, default True
+        sort : False or None, default None
             Sort the resulting MultiIndex if possible
 
             .. versionadded:: 0.24.0
 
+            .. versionchanged:: 0.24.1
+
+               Changed the default value from ``True`` to ``None``
+               (without change in behaviour).
+
         Returns
         -------
         diff : MultiIndex
         """
+        self._validate_sort_keyword(sort)
         self._assert_can_do_setop(other)
         other, result_names = self._convert_can_do_setop(other)
 
@@ -2986,7 +3011,7 @@ def difference(self, other, sort=True):
         label_diff = np.setdiff1d(np.arange(this.size), indexer,
                                   assume_unique=True)
         difference = this.values.take(label_diff)
-        if sort:
+        if sort is None:
             difference = sorted(difference)
 
         if len(difference) == 0:
diff --git a/pandas/core/indexes/range.py b/pandas/core/indexes/range.py
index e17a6a682af40..5aafe9734b6a0 100644
--- a/pandas/core/indexes/range.py
+++ b/pandas/core/indexes/range.py
@@ -350,19 +350,21 @@ def intersection(self, other, sort=False):
         Parameters
         ----------
         other : Index or array-like
-        sort : bool, default True
+        sort : False or None, default False
             Sort the resulting index if possible
 
             .. versionadded:: 0.24.0
 
             .. versionchanged:: 0.24.1
 
-               Changed the default from ``True`` to ``False``.
+               Changed the default to ``False`` to match the behaviour
+               from before 0.24.0.
 
         Returns
         -------
         intersection : Index
         """
+        self._validate_sort_keyword(sort)
 
         if self.equals(other):
             return self._get_reconciled_name_object(other)
@@ -405,7 +407,7 @@ def intersection(self, other, sort=False):
 
         if (self._step < 0 and other._step < 0) is not (new_index._step < 0):
             new_index = new_index[::-1]
-        if sort:
+        if sort is None:
             new_index = new_index.sort_values()
         return new_index
 
diff --git a/pandas/tests/indexes/common.py b/pandas/tests/indexes/common.py
index d72dccadf0ac0..a838779689c44 100644
--- a/pandas/tests/indexes/common.py
+++ b/pandas/tests/indexes/common.py
@@ -486,7 +486,7 @@ def test_union_base(self):
                 with pytest.raises(TypeError, match=msg):
                     first.union([1, 2, 3])
 
-    @pytest.mark.parametrize("sort", [True, False])
+    @pytest.mark.parametrize("sort", [None, False])
     def test_difference_base(self, sort):
         for name, idx in compat.iteritems(self.indices):
             first = idx[2:]
diff --git a/pandas/tests/indexes/datetimes/test_setops.py b/pandas/tests/indexes/datetimes/test_setops.py
index bd37cc815d0f7..19009e45ee83a 100644
--- a/pandas/tests/indexes/datetimes/test_setops.py
+++ b/pandas/tests/indexes/datetimes/test_setops.py
@@ -138,7 +138,7 @@ def test_intersection2(self):
 
     @pytest.mark.parametrize("tz", [None, 'Asia/Tokyo', 'US/Eastern',
                                     'dateutil/US/Pacific'])
-    @pytest.mark.parametrize("sort", [True, False])
+    @pytest.mark.parametrize("sort", [None, False])
     def test_intersection(self, tz, sort):
         # GH 4690 (with tz)
         base = date_range('6/1/2000', '6/30/2000', freq='D', name='idx')
@@ -187,7 +187,7 @@ def test_intersection(self, tz, sort):
         for (rng, expected) in [(rng2, expected2), (rng3, expected3),
                                 (rng4, expected4)]:
             result = base.intersection(rng, sort=sort)
-            if sort:
+            if sort is None:
                 expected = expected.sort_values()
             tm.assert_index_equal(result, expected)
             assert result.name == expected.name
@@ -212,7 +212,7 @@ def test_intersection_bug_1708(self):
         assert len(result) == 0
 
     @pytest.mark.parametrize("tz", tz)
-    @pytest.mark.parametrize("sort", [True, False])
+    @pytest.mark.parametrize("sort", [None, False])
     def test_difference(self, tz, sort):
         rng_dates = ['1/2/2000', '1/3/2000', '1/1/2000', '1/4/2000',
                      '1/5/2000']
@@ -233,11 +233,11 @@ def test_difference(self, tz, sort):
                                      (rng2, other2, expected2),
                                      (rng3, other3, expected3)]:
             result_diff = rng.difference(other, sort)
-            if sort:
+            if sort is None:
                 expected = expected.sort_values()
             tm.assert_index_equal(result_diff, expected)
 
-    @pytest.mark.parametrize("sort", [True, False])
+    @pytest.mark.parametrize("sort", [None, False])
     def test_difference_freq(self, sort):
         # GH14323: difference of DatetimeIndex should not preserve frequency
 
@@ -254,7 +254,7 @@ def test_difference_freq(self, sort):
         tm.assert_index_equal(idx_diff, expected)
         tm.assert_attr_equal('freq', idx_diff, expected)
 
-    @pytest.mark.parametrize("sort", [True, False])
+    @pytest.mark.parametrize("sort", [None, False])
     def test_datetimeindex_diff(self, sort):
         dti1 = date_range(freq='Q-JAN', start=datetime(1997, 12, 31),
                           periods=100)
diff --git a/pandas/tests/indexes/interval/test_interval.py b/pandas/tests/indexes/interval/test_interval.py
index db69258c1d3d2..f1fd06c9cab6e 100644
--- a/pandas/tests/indexes/interval/test_interval.py
+++ b/pandas/tests/indexes/interval/test_interval.py
@@ -783,19 +783,19 @@ def test_non_contiguous(self, closed):
 
         assert 1.5 not in index
 
-    @pytest.mark.parametrize("sort", [True, False])
+    @pytest.mark.parametrize("sort", [None, False])
     def test_union(self, closed, sort):
         index = self.create_index(closed=closed)
         other = IntervalIndex.from_breaks(range(5, 13), closed=closed)
 
         expected = IntervalIndex.from_breaks(range(13), closed=closed)
         result = index[::-1].union(other, sort=sort)
-        if sort:
+        if sort is None:
             tm.assert_index_equal(result, expected)
         assert tm.equalContents(result, expected)
 
         result = other[::-1].union(index, sort=sort)
-        if sort:
+        if sort is None:
             tm.assert_index_equal(result, expected)
         assert tm.equalContents(result, expected)
 
@@ -812,19 +812,19 @@ def test_union(self, closed, sort):
         result = index.union(other, sort=sort)
         tm.assert_index_equal(result, index)
 
-    @pytest.mark.parametrize("sort", [True, False])
+    @pytest.mark.parametrize("sort", [None, False])
     def test_intersection(self, closed, sort):
         index = self.create_index(closed=closed)
         other = IntervalIndex.from_breaks(range(5, 13), closed=closed)
 
         expected = IntervalIndex.from_breaks(range(5, 11), closed=closed)
         result = index[::-1].intersection(other, sort=sort)
-        if sort:
+        if sort is None:
             tm.assert_index_equal(result, expected)
         assert tm.equalContents(result, expected)
 
         result = other[::-1].intersection(index, sort=sort)
-        if sort:
+        if sort is None:
             tm.assert_index_equal(result, expected)
         assert tm.equalContents(result, expected)
 
@@ -842,14 +842,14 @@ def test_intersection(self, closed, sort):
         result = index.intersection(other, sort=sort)
         tm.assert_index_equal(result, expected)
 
-    @pytest.mark.parametrize("sort", [True, False])
+    @pytest.mark.parametrize("sort", [None, False])
     def test_difference(self, closed, sort):
         index = IntervalIndex.from_arrays([1, 0, 3, 2],
                                           [1, 2, 3, 4],
                                           closed=closed)
         result = index.difference(index[:1], sort=sort)
         expected = index[1:]
-        if sort:
+        if sort is None:
             expected = expected.sort_values()
         tm.assert_index_equal(result, expected)
 
@@ -864,19 +864,19 @@ def test_difference(self, closed, sort):
         result = index.difference(other, sort=sort)
         tm.assert_index_equal(result, expected)
 
-    @pytest.mark.parametrize("sort", [True, False])
+    @pytest.mark.parametrize("sort", [None, False])
     def test_symmetric_difference(self, closed, sort):
         index = self.create_index(closed=closed)
         result = index[1:].symmetric_difference(index[:-1], sort=sort)
         expected = IntervalIndex([index[0], index[-1]])
-        if sort:
+        if sort is None:
             tm.assert_index_equal(result, expected)
         assert tm.equalContents(result, expected)
 
         # GH 19101: empty result, same dtype
         result = index.symmetric_difference(index, sort=sort)
         expected = IntervalIndex(np.array([], dtype='int64'), closed=closed)
-        if sort:
+        if sort is None:
             tm.assert_index_equal(result, expected)
         assert tm.equalContents(result, expected)
 
@@ -888,7 +888,7 @@ def test_symmetric_difference(self, closed, sort):
 
     @pytest.mark.parametrize('op_name', [
         'union', 'intersection', 'difference', 'symmetric_difference'])
-    @pytest.mark.parametrize("sort", [True, False])
+    @pytest.mark.parametrize("sort", [None, False])
     def test_set_operation_errors(self, closed, op_name, sort):
         index = self.create_index(closed=closed)
         set_op = getattr(index, op_name)
diff --git a/pandas/tests/indexes/multi/test_set_ops.py b/pandas/tests/indexes/multi/test_set_ops.py
index 208d6cf1c639f..41a0e1e59e8a5 100644
--- a/pandas/tests/indexes/multi/test_set_ops.py
+++ b/pandas/tests/indexes/multi/test_set_ops.py
@@ -9,7 +9,7 @@
 
 
 @pytest.mark.parametrize("case", [0.5, "xxx"])
-@pytest.mark.parametrize("sort", [True, False])
+@pytest.mark.parametrize("sort", [None, False])
 @pytest.mark.parametrize("method", ["intersection", "union",
                                     "difference", "symmetric_difference"])
 def test_set_ops_error_cases(idx, case, sort, method):
@@ -19,13 +19,13 @@ def test_set_ops_error_cases(idx, case, sort, method):
         getattr(idx, method)(case, sort=sort)
 
 
-@pytest.mark.parametrize("sort", [True, False])
+@pytest.mark.parametrize("sort", [None, False])
 def test_intersection_base(idx, sort):
     first = idx[:5]
     second = idx[:3]
     intersect = first.intersection(second, sort=sort)
 
-    if sort:
+    if sort is None:
         tm.assert_index_equal(intersect, second.sort_values())
     assert tm.equalContents(intersect, second)
 
@@ -34,7 +34,7 @@ def test_intersection_base(idx, sort):
              for klass in [np.array, Series, list]]
     for case in cases:
         result = first.intersection(case, sort=sort)
-        if sort:
+        if sort is None:
             tm.assert_index_equal(result, second.sort_values())
         assert tm.equalContents(result, second)
 
@@ -43,13 +43,13 @@ def test_intersection_base(idx, sort):
         first.intersection([1, 2, 3], sort=sort)
 
 
-@pytest.mark.parametrize("sort", [True, False])
+@pytest.mark.parametrize("sort", [None, False])
 def test_union_base(idx, sort):
     first = idx[3:]
     second = idx[:5]
     everything = idx
     union = first.union(second, sort=sort)
-    if sort:
+    if sort is None:
         tm.assert_index_equal(union, everything.sort_values())
     assert tm.equalContents(union, everything)
 
@@ -58,7 +58,7 @@ def test_union_base(idx, sort):
              for klass in [np.array, Series, list]]
     for case in cases:
         result = first.union(case, sort=sort)
-        if sort:
+        if sort is None:
             tm.assert_index_equal(result, everything.sort_values())
         assert tm.equalContents(result, everything)
 
@@ -67,13 +67,13 @@ def test_union_base(idx, sort):
         first.union([1, 2, 3], sort=sort)
 
 
-@pytest.mark.parametrize("sort", [True, False])
+@pytest.mark.parametrize("sort", [None, False])
 def test_difference_base(idx, sort):
     second = idx[4:]
     answer = idx[:4]
     result = idx.difference(second, sort=sort)
 
-    if sort:
+    if sort is None:
         answer = answer.sort_values()
 
     assert result.equals(answer)
@@ -91,14 +91,14 @@ def test_difference_base(idx, sort):
         idx.difference([1, 2, 3], sort=sort)
 
 
-@pytest.mark.parametrize("sort", [True, False])
+@pytest.mark.parametrize("sort", [None, False])
 def test_symmetric_difference(idx, sort):
     first = idx[1:]
     second = idx[:-1]
     answer = idx[[-1, 0]]
     result = first.symmetric_difference(second, sort=sort)
 
-    if sort:
+    if sort is None:
         answer = answer.sort_values()
 
     tm.assert_index_equal(result, answer)
@@ -121,14 +121,14 @@ def test_empty(idx):
     assert idx[:0].empty
 
 
-@pytest.mark.parametrize("sort", [True, False])
+@pytest.mark.parametrize("sort", [None, False])
 def test_difference(idx, sort):
 
     first = idx
     result = first.difference(idx[-3:], sort=sort)
     vals = idx[:-3].values
 
-    if sort:
+    if sort is None:
         vals = sorted(vals)
 
     expected = MultiIndex.from_tuples(vals,
@@ -189,14 +189,62 @@ def test_difference(idx, sort):
         first.difference([1, 2, 3, 4, 5], sort=sort)
 
 
-@pytest.mark.parametrize("sort", [True, False])
+def test_difference_sort_special():
+    # GH-24959
+    idx = pd.MultiIndex.from_product([[1, 0], ['a', 'b']])
+    # sort=None, the default
+    result = idx.difference([])
+    tm.assert_index_equal(result, idx)
+
+
+@pytest.mark.xfail(reason="Not implemented.")
+def test_difference_sort_special_true():
+    # TODO decide on True behaviour
+    idx = pd.MultiIndex.from_product([[1, 0], ['a', 'b']])
+    result = idx.difference([], sort=True)
+    expected = pd.MultiIndex.from_product([[0, 1], ['a', 'b']])
+    tm.assert_index_equal(result, expected)
+
+
+def test_difference_sort_incomparable():
+    # GH-24959
+    idx = pd.MultiIndex.from_product([[1, pd.Timestamp('2000'), 2],
+                                      ['a', 'b']])
+
+    other = pd.MultiIndex.from_product([[3, pd.Timestamp('2000'), 4],
+                                        ['c', 'd']])
+    # sort=None, the default
+    # MultiIndex.difference deviates here from other difference
+    # implementations in not catching the TypeError
+    with pytest.raises(TypeError):
+        result = idx.difference(other)
+
+    # sort=False
+    result = idx.difference(other, sort=False)
+    tm.assert_index_equal(result, idx)
+
+
+@pytest.mark.xfail(reason="Not implemented.")
+def test_difference_sort_incomparable_true():
+    # TODO decide on True behaviour
+    # # sort=True, raises
+    idx = pd.MultiIndex.from_product([[1, pd.Timestamp('2000'), 2],
+                                      ['a', 'b']])
+    other = pd.MultiIndex.from_product([[3, pd.Timestamp('2000'), 4],
+                                        ['c', 'd']])
+
+    with pytest.raises(TypeError):
+        idx.difference(other, sort=True)
+
+
+@pytest.mark.parametrize("sort", [None, False])
 def test_union(idx, sort):
     piece1 = idx[:5][::-1]
     piece2 = idx[3:]
 
     the_union = piece1.union(piece2, sort=sort)
 
-    if sort:
+    if sort is None:
         tm.assert_index_equal(the_union, idx.sort_values())
 
     assert tm.equalContents(the_union, idx)
@@ -225,14 +273,14 @@ def test_union(idx, sort):
     #     assert result.equals(result2)
 
 
-@pytest.mark.parametrize("sort", [True, False])
+@pytest.mark.parametrize("sort", [None, False])
 def test_intersection(idx, sort):
     piece1 = idx[:5][::-1]
     piece2 = idx[3:]
 
     the_int = piece1.intersection(piece2, sort=sort)
 
-    if sort:
+    if sort is None:
         tm.assert_index_equal(the_int, idx[3:5])
     assert tm.equalContents(the_int, idx[3:5])
 
@@ -249,3 +297,76 @@ def test_intersection(idx, sort):
     # tuples = _index.values
     # result = _index & tuples
     # assert result.equals(tuples)
+
+
+def test_intersect_equal_sort():
+    # GH-24959
+    idx = pd.MultiIndex.from_product([[1, 0], ['a', 'b']])
+    tm.assert_index_equal(idx.intersection(idx, sort=False), idx)
+    tm.assert_index_equal(idx.intersection(idx, sort=None), idx)
+
+
+@pytest.mark.xfail(reason="Not implemented.")
+def test_intersect_equal_sort_true():
+    # TODO decide on True behaviour
+    idx = pd.MultiIndex.from_product([[1, 0], ['a', 'b']])
+    sorted_ = pd.MultiIndex.from_product([[0, 1], ['a', 'b']])
+    tm.assert_index_equal(idx.intersection(idx, sort=True), sorted_)
+
+
+@pytest.mark.parametrize('slice_', [slice(None), slice(0)])
+def test_union_sort_other_empty(slice_):
+    # https://github.com/pandas-dev/pandas/issues/24959
+    idx = pd.MultiIndex.from_product([[1, 0], ['a', 'b']])
+
+    # default, sort=None
+    other = idx[slice_]
+    tm.assert_index_equal(idx.union(other), idx)
+    # MultiIndex does not special case empty.union(idx)
+    # tm.assert_index_equal(other.union(idx), idx)
+
+    # sort=False
+    tm.assert_index_equal(idx.union(other, sort=False), idx)
+
+
+@pytest.mark.xfail(reason="Not implemented.")
+def test_union_sort_other_empty_sort(slice_):
+    # TODO decide on True behaviour
+    # # sort=True
+    idx = pd.MultiIndex.from_product([[1, 0], ['a', 'b']])
+    other = idx[:0]
+    result = idx.union(other, sort=True)
+    expected = pd.MultiIndex.from_product([[0, 1], ['a', 'b']])
+    tm.assert_index_equal(result, expected)
+
+
+def test_union_sort_other_incomparable():
+    # https://github.com/pandas-dev/pandas/issues/24959
+    idx = pd.MultiIndex.from_product([[1, pd.Timestamp('2000')], ['a', 'b']])
+
+    # default, sort=None
+    result = idx.union(idx[:1])
+    tm.assert_index_equal(result, idx)
+
+    # sort=False
+    result = idx.union(idx[:1], sort=False)
+    tm.assert_index_equal(result, idx)
+
+
+@pytest.mark.xfail(reason="Not implemented.")
+def test_union_sort_other_incomparable_sort():
+    # TODO decide on True behaviour
+    # # sort=True
+    idx = pd.MultiIndex.from_product([[1, pd.Timestamp('2000')], ['a', 'b']])
+    with pytest.raises(TypeError, match='Cannot compare'):
+        idx.union(idx[:1], sort=True)
+
+
+@pytest.mark.parametrize("method", ['union', 'intersection', 'difference',
+                                    'symmetric_difference'])
+def test_setops_disallow_true(method):
+    idx1 = pd.MultiIndex.from_product([['a', 'b'], [1, 2]])
+    idx2 = pd.MultiIndex.from_product([['b', 'c'], [1, 2]])
+
+    with pytest.raises(ValueError, match="The 'sort' keyword only takes"):
+        getattr(idx1, method)(idx2, sort=True)
diff --git a/pandas/tests/indexes/period/test_period.py b/pandas/tests/indexes/period/test_period.py
index 464ff7aa5d58d..dc9a32d75d272 100644
--- a/pandas/tests/indexes/period/test_period.py
+++ b/pandas/tests/indexes/period/test_period.py
@@ -77,7 +77,7 @@ def test_no_millisecond_field(self):
         with pytest.raises(AttributeError):
             DatetimeIndex([]).millisecond
 
-    @pytest.mark.parametrize("sort", [True, False])
+    @pytest.mark.parametrize("sort", [None, False])
     def test_difference_freq(self, sort):
         # GH14323: difference of Period MUST preserve frequency
         # but the ability to union results must be preserved
diff --git a/pandas/tests/indexes/period/test_setops.py b/pandas/tests/indexes/period/test_setops.py
index a97ab47bcda16..bf29edad4841e 100644
--- a/pandas/tests/indexes/period/test_setops.py
+++ b/pandas/tests/indexes/period/test_setops.py
@@ -38,7 +38,7 @@ def test_join_does_not_recur(self):
                           df.columns[0], df.columns[1]], object)
         tm.assert_index_equal(res, expected)
 
-    @pytest.mark.parametrize("sort", [True, False])
+    @pytest.mark.parametrize("sort", [None, False])
     def test_union(self, sort):
         # union
         other1 = pd.period_range('1/1/2000', freq='D', periods=5)
@@ -97,11 +97,11 @@ def test_union(self, sort):
                                      (rng8, other8, expected8)]:
 
             result_union = rng.union(other, sort=sort)
-            if sort:
+            if sort is None:
                 expected = expected.sort_values()
             tm.assert_index_equal(result_union, expected)
 
-    @pytest.mark.parametrize("sort", [True, False])
+    @pytest.mark.parametrize("sort", [None, False])
     def test_union_misc(self, sort):
         index = period_range('1/1/2000', '1/20/2000', freq='D')
 
@@ -110,7 +110,7 @@ def test_union_misc(self, sort):
 
         # not in order
         result = _permute(index[:-5]).union(_permute(index[10:]), sort=sort)
-        if sort:
+        if sort is None:
             tm.assert_index_equal(result, index)
         assert tm.equalContents(result, index)
 
@@ -139,7 +139,7 @@ def test_union_dataframe_index(self):
         exp = pd.period_range('1/1/1980', '1/1/2012', freq='M')
         tm.assert_index_equal(df.index, exp)
 
-    @pytest.mark.parametrize("sort", [True, False])
+    @pytest.mark.parametrize("sort", [None, False])
     def test_intersection(self, sort):
         index = period_range('1/1/2000', '1/20/2000', freq='D')
 
@@ -150,7 +150,7 @@ def test_intersection(self, sort):
         left = _permute(index[:-5])
         right = _permute(index[10:])
         result = left.intersection(right, sort=sort)
-        if sort:
+        if sort is None:
             tm.assert_index_equal(result, index[10:-5])
         assert tm.equalContents(result, index[10:-5])
 
@@ -164,7 +164,7 @@ def test_intersection(self, sort):
         with pytest.raises(period.IncompatibleFrequency):
             index.intersection(index3, sort=sort)
 
-    @pytest.mark.parametrize("sort", [True, False])
+    @pytest.mark.parametrize("sort", [None, False])
     def test_intersection_cases(self, sort):
         base = period_range('6/1/2000', '6/30/2000', freq='D', name='idx')
 
@@ -210,7 +210,7 @@ def test_intersection_cases(self, sort):
         for (rng, expected) in [(rng2, expected2), (rng3, expected3),
                                 (rng4, expected4)]:
             result = base.intersection(rng, sort=sort)
-            if sort:
+            if sort is None:
                 expected = expected.sort_values()
             tm.assert_index_equal(result, expected)
             assert result.name == expected.name
@@ -224,7 +224,7 @@ def test_intersection_cases(self, sort):
         result = rng.intersection(rng[0:0])
         assert len(result) == 0
 
-    @pytest.mark.parametrize("sort", [True, False])
+    @pytest.mark.parametrize("sort", [None, False])
     def test_difference(self, sort):
         # diff
         period_rng = ['1/3/2000', '1/2/2000', '1/1/2000', '1/5/2000',
@@ -276,6 +276,6 @@ def test_difference(self, sort):
                                      (rng6, other6, expected6),
                                      (rng7, other7, expected7), ]:
             result_difference = rng.difference(other, sort=sort)
-            if sort:
+            if sort is None:
                 expected = expected.sort_values()
             tm.assert_index_equal(result_difference, expected)
diff --git a/pandas/tests/indexes/test_base.py b/pandas/tests/indexes/test_base.py
index 20e439de46bde..c99007cef90d4 100644
--- a/pandas/tests/indexes/test_base.py
+++ b/pandas/tests/indexes/test_base.py
@@ -3,6 +3,7 @@
 from collections import defaultdict
 from datetime import datetime, timedelta
 import math
+import operator
 import sys
 
 import numpy as np
@@ -684,12 +685,12 @@ def test_empty_fancy_raises(self, attr):
         # np.ndarray only accepts ndarray of int & bool dtypes, so should Index
         pytest.raises(IndexError, index.__getitem__, empty_farr)
 
-    @pytest.mark.parametrize("sort", [True, False])
+    @pytest.mark.parametrize("sort", [None, False])
     def test_intersection(self, sort):
         first = self.strIndex[:20]
         second = self.strIndex[:10]
         intersect = first.intersection(second, sort=sort)
-        if sort:
+        if sort is None:
             tm.assert_index_equal(intersect, second.sort_values())
         assert tm.equalContents(intersect, second)
 
@@ -701,7 +702,7 @@ def test_intersection(self, sort):
         (Index([3, 4, 5, 6, 7], name="index"), True),  # preserve same name
         (Index([3, 4, 5, 6, 7], name="other"), False),  # drop diff names
         (Index([3, 4, 5, 6, 7]), False)])
-    @pytest.mark.parametrize("sort", [True, False])
+    @pytest.mark.parametrize("sort", [None, False])
     def test_intersection_name_preservation(self, index2, keeps_name, sort):
         index1 = Index([1, 2, 3, 4, 5], name='index')
         expected = Index([3, 4, 5])
@@ -715,7 +716,7 @@ def test_intersection_name_preservation(self, index2, keeps_name, sort):
 
     @pytest.mark.parametrize("first_name,second_name,expected_name", [
         ('A', 'A', 'A'), ('A', 'B', None), (None, 'B', None)])
-    @pytest.mark.parametrize("sort", [True, False])
+    @pytest.mark.parametrize("sort", [None, False])
     def test_intersection_name_preservation2(self, first_name, second_name,
                                              expected_name, sort):
         first = self.strIndex[5:20]
@@ -728,7 +729,7 @@ def test_intersection_name_preservation2(self, first_name, second_name,
     @pytest.mark.parametrize("index2,keeps_name", [
         (Index([4, 7, 6, 5, 3], name='index'), True),
         (Index([4, 7, 6, 5, 3], name='other'), False)])
-    @pytest.mark.parametrize("sort", [True, False])
+    @pytest.mark.parametrize("sort", [None, False])
     def test_intersection_monotonic(self, index2, keeps_name, sort):
         index1 = Index([5, 3, 2, 4, 1], name='index')
         expected = Index([5, 3, 4])
@@ -737,25 +738,25 @@ def test_intersection_monotonic(self, index2, keeps_name, sort):
             expected.name = "index"
 
         result = index1.intersection(index2, sort=sort)
-        if sort:
+        if sort is None:
             expected = expected.sort_values()
         tm.assert_index_equal(result, expected)
 
     @pytest.mark.parametrize("index2,expected_arr", [
         (Index(['B', 'D']), ['B']),
         (Index(['B', 'D', 'A']), ['A', 'B', 'A'])])
-    @pytest.mark.parametrize("sort", [True, False])
+    @pytest.mark.parametrize("sort", [None, False])
     def test_intersection_non_monotonic_non_unique(self, index2, expected_arr,
                                                    sort):
         # non-monotonic non-unique
         index1 = Index(['A', 'B', 'A', 'C'])
         expected = Index(expected_arr, dtype='object')
         result = index1.intersection(index2, sort=sort)
-        if sort:
+        if sort is None:
             expected = expected.sort_values()
         tm.assert_index_equal(result, expected)
 
-    @pytest.mark.parametrize("sort", [True, False])
+    @pytest.mark.parametrize("sort", [None, False])
     def test_intersect_str_dates(self, sort):
         dt_dates = [datetime(2012, 2, 9), datetime(2012, 2, 22)]
 
@@ -770,7 +771,19 @@ def test_intersect_nosort(self):
         expected = pd.Index(['b', 'a'])
         tm.assert_index_equal(result, expected)
 
-    @pytest.mark.parametrize("sort", [True, False])
+    def test_intersection_equal_sort(self):
+        idx = pd.Index(['c', 'a', 'b'])
+        tm.assert_index_equal(idx.intersection(idx, sort=False), idx)
+        tm.assert_index_equal(idx.intersection(idx, sort=None), idx)
+
+    @pytest.mark.xfail(reason="Not implemented")
+    def test_intersection_equal_sort_true(self):
+        # TODO decide on True behaviour
+        idx = pd.Index(['c', 'a', 'b'])
+        sorted_ = pd.Index(['a', 'b', 'c'])
+        tm.assert_index_equal(idx.intersection(idx, sort=True), sorted_)
+
+    @pytest.mark.parametrize("sort", [None, False])
     def test_chained_union(self, sort):
         # Chained unions handles names correctly
         i1 = Index([1, 2], name='i1')
@@ -787,7 +800,7 @@ def test_chained_union(self, sort):
         expected = j1.union(j2, sort=sort).union(j3, sort=sort)
         tm.assert_index_equal(union, expected)
 
-    @pytest.mark.parametrize("sort", [True, False])
+    @pytest.mark.parametrize("sort", [None, False])
     def test_union(self, sort):
         # TODO: Replace with fixturesult
         first = self.strIndex[5:20]
@@ -795,13 +808,65 @@ def test_union(self, sort):
         everything = self.strIndex[:20]
 
         union = first.union(second, sort=sort)
-        if sort:
+        if sort is None:
             tm.assert_index_equal(union, everything.sort_values())
         assert tm.equalContents(union, everything)
 
+    @pytest.mark.parametrize('slice_', [slice(None), slice(0)])
+    def test_union_sort_other_special(self, slice_):
+        # https://github.com/pandas-dev/pandas/issues/24959
+
+        idx = pd.Index([1, 0, 2])
+        # default, sort=None
+        other = idx[slice_]
+        tm.assert_index_equal(idx.union(other), idx)
+        tm.assert_index_equal(other.union(idx), idx)
+
+        # sort=False
+        tm.assert_index_equal(idx.union(other, sort=False), idx)
+
+    @pytest.mark.xfail(reason="Not implemented")
+    @pytest.mark.parametrize('slice_', [slice(None), slice(0)])
+    def test_union_sort_special_true(self, slice_):
+        # TODO decide on True behaviour
+        # sort=True
+        idx = pd.Index([1, 0, 2])
+        # default, sort=None
+        other = idx[slice_]
+
+        result = idx.union(other, sort=True)
+        expected = pd.Index([0, 1, 2])
+        tm.assert_index_equal(result, expected)
+
+    def test_union_sort_other_incomparable(self):
+        # https://github.com/pandas-dev/pandas/issues/24959
+        idx = pd.Index([1, pd.Timestamp('2000')])
+        # default (sort=None)
+        with tm.assert_produces_warning(RuntimeWarning):
+            result = idx.union(idx[:1])
+
+        tm.assert_index_equal(result, idx)
+
+        # sort=None
+        with tm.assert_produces_warning(RuntimeWarning):
+            result = idx.union(idx[:1], sort=None)
+        tm.assert_index_equal(result, idx)
+
+        # sort=False
+        result = idx.union(idx[:1], sort=False)
+        tm.assert_index_equal(result, idx)
+
+    @pytest.mark.xfail(reason="Not implemented")
+    def test_union_sort_other_incomparable_true(self):
+        # TODO decide on True behaviour
+        # sort=True
+        idx = pd.Index([1, pd.Timestamp('2000')])
+        with pytest.raises(TypeError, match='.*'):
+            idx.union(idx[:1], sort=True)
+
     @pytest.mark.parametrize("klass", [
         np.array, Series, list])
-    @pytest.mark.parametrize("sort", [True, False])
+    @pytest.mark.parametrize("sort", [None, False])
     def test_union_from_iterables(self, klass, sort):
         # GH 10149
         # TODO: Replace with fixturesult
@@ -811,29 +876,30 @@ def test_union_from_iterables(self, klass, sort):
 
         case = klass(second.values)
         result = first.union(case, sort=sort)
-        if sort:
+        if sort is None:
             tm.assert_index_equal(result, everything.sort_values())
         assert tm.equalContents(result, everything)
 
-    @pytest.mark.parametrize("sort", [True, False])
+    @pytest.mark.parametrize("sort", [None, False])
     def test_union_identity(self, sort):
         # TODO: replace with fixturesult
         first = self.strIndex[5:20]
 
         union = first.union(first, sort=sort)
-        assert union is first
+        # i.e. identity is not preserved when sort is True
+        assert (union is first) is (not sort)
 
         union = first.union([], sort=sort)
-        assert union is first
+        assert (union is first) is (not sort)
 
         union = Index([]).union(first, sort=sort)
-        assert union is first
+        assert (union is first) is (not sort)
 
     @pytest.mark.parametrize("first_list", [list('ba'), list()])
     @pytest.mark.parametrize("second_list", [list('ab'), list()])
     @pytest.mark.parametrize("first_name, second_name, expected_name", [
         ('A', 'B', None), (None, 'B', None), ('A', None, None)])
-    @pytest.mark.parametrize("sort", [True, False])
+    @pytest.mark.parametrize("sort", [None, False])
     def test_union_name_preservation(self, first_list, second_list, first_name,
                                      second_name, expected_name, sort):
         first = Index(first_list, name=first_name)
@@ -842,14 +908,14 @@ def test_union_name_preservation(self, first_list, second_list, first_name,
 
         vals = set(first_list).union(second_list)
 
-        if sort and len(first_list) > 0 and len(second_list) > 0:
+        if sort is None and len(first_list) > 0 and len(second_list) > 0:
             expected = Index(sorted(vals), name=expected_name)
             tm.assert_index_equal(union, expected)
         else:
             expected = Index(vals, name=expected_name)
             assert tm.equalContents(union, expected)
 
-    @pytest.mark.parametrize("sort", [True, False])
+    @pytest.mark.parametrize("sort", [None, False])
     def test_union_dt_as_obj(self, sort):
         # TODO: Replace with fixturesult
         firstCat = self.strIndex.union(self.dateIndex)
@@ -866,6 +932,15 @@ def test_union_dt_as_obj(self, sort):
         tm.assert_contains_all(self.strIndex, secondCat)
         tm.assert_contains_all(self.dateIndex, firstCat)
 
+    @pytest.mark.parametrize("method", ['union', 'intersection', 'difference',
+                                        'symmetric_difference'])
+    def test_setops_disallow_true(self, method):
+        idx1 = pd.Index(['a', 'b'])
+        idx2 = pd.Index(['b', 'c'])
+
+        with pytest.raises(ValueError, match="The 'sort' keyword only takes"):
+            getattr(idx1, method)(idx2, sort=True)
+
     def test_map_identity_mapping(self):
         # GH 12766
         # TODO: replace with fixture
@@ -987,7 +1062,7 @@ def test_append_empty_preserve_name(self, name, expected):
 
     @pytest.mark.parametrize("second_name,expected", [
         (None, None), ('name', 'name')])
-    @pytest.mark.parametrize("sort", [True, False])
+    @pytest.mark.parametrize("sort", [None, False])
     def test_difference_name_preservation(self, second_name, expected, sort):
         # TODO: replace with fixturesult
         first = self.strIndex[5:20]
@@ -1005,7 +1080,7 @@ def test_difference_name_preservation(self, second_name, expected, sort):
         else:
             assert result.name == expected
 
-    @pytest.mark.parametrize("sort", [True, False])
+    @pytest.mark.parametrize("sort", [None, False])
     def test_difference_empty_arg(self, sort):
         first = self.strIndex[5:20]
         first.name == 'name'
@@ -1014,7 +1089,7 @@ def test_difference_empty_arg(self, sort):
         assert tm.equalContents(result, first)
         assert result.name == first.name
 
-    @pytest.mark.parametrize("sort", [True, False])
+    @pytest.mark.parametrize("sort", [None, False])
     def test_difference_identity(self, sort):
         first = self.strIndex[5:20]
         first.name == 'name'
@@ -1023,7 +1098,7 @@ def test_difference_identity(self, sort):
         assert len(result) == 0
         assert result.name == first.name
 
-    @pytest.mark.parametrize("sort", [True, False])
+    @pytest.mark.parametrize("sort", [None, False])
     def test_difference_sort(self, sort):
         first = self.strIndex[5:20]
         second = self.strIndex[:10]
@@ -1031,12 +1106,12 @@ def test_difference_sort(self, sort):
         result = first.difference(second, sort)
         expected = self.strIndex[10:20]
 
-        if sort:
+        if sort is None:
             expected = expected.sort_values()
 
         tm.assert_index_equal(result, expected)
 
-    @pytest.mark.parametrize("sort", [True, False])
+    @pytest.mark.parametrize("sort", [None, False])
     def test_symmetric_difference(self, sort):
         # smoke
         index1 = Index([5, 2, 3, 4], name='index1')
@@ -1045,7 +1120,7 @@ def test_symmetric_difference(self, sort):
         expected = Index([5, 1])
         assert tm.equalContents(result, expected)
         assert result.name is None
-        if sort:
+        if sort is None:
             expected = expected.sort_values()
         tm.assert_index_equal(result, expected)
 
@@ -1054,13 +1129,43 @@ def test_symmetric_difference(self, sort):
         assert tm.equalContents(result, expected)
         assert result.name is None
 
-    @pytest.mark.parametrize("sort", [True, False])
+    @pytest.mark.parametrize('opname', ['difference', 'symmetric_difference'])
+    def test_difference_incomparable(self, opname):
+        a = pd.Index([3, pd.Timestamp('2000'), 1])
+        b = pd.Index([2, pd.Timestamp('1999'), 1])
+        op = operator.methodcaller(opname, b)
+
+        # sort=None, the default
+        result = op(a)
+        expected = pd.Index([3, pd.Timestamp('2000'), 2, pd.Timestamp('1999')])
+        if opname == 'difference':
+            expected = expected[:2]
+        tm.assert_index_equal(result, expected)
+
+        # sort=False
+        op = operator.methodcaller(opname, b, sort=False)
+        result = op(a)
+        tm.assert_index_equal(result, expected)
+
+    @pytest.mark.xfail(reason="Not implemented")
+    @pytest.mark.parametrize('opname', ['difference', 'symmetric_difference'])
+    def test_difference_incomparable_true(self, opname):
+        # TODO decide on True behaviour
+        # # sort=True, raises
+        a = pd.Index([3, pd.Timestamp('2000'), 1])
+        b = pd.Index([2, pd.Timestamp('1999'), 1])
+        op = operator.methodcaller(opname, b, sort=True)
+
+        with pytest.raises(TypeError, match='Cannot compare'):
+            op(a)
+
+    @pytest.mark.parametrize("sort", [None, False])
     def test_symmetric_difference_mi(self, sort):
         index1 = MultiIndex.from_tuples(self.tuples)
         index2 = MultiIndex.from_tuples([('foo', 1), ('bar', 3)])
         result = index1.symmetric_difference(index2, sort=sort)
         expected = MultiIndex.from_tuples([('bar', 2), ('baz', 3), ('bar', 3)])
-        if sort:
+        if sort is None:
             expected = expected.sort_values()
         tm.assert_index_equal(result, expected)
         assert tm.equalContents(result, expected)
@@ -1068,18 +1173,18 @@ def test_symmetric_difference_mi(self, sort):
     @pytest.mark.parametrize("index2,expected", [
         (Index([0, 1, np.nan]), Index([2.0, 3.0, 0.0])),
         (Index([0, 1]), Index([np.nan, 2.0, 3.0, 0.0]))])
-    @pytest.mark.parametrize("sort", [True, False])
+    @pytest.mark.parametrize("sort", [None, False])
     def test_symmetric_difference_missing(self, index2, expected, sort):
         # GH 13514 change: {nan} - {nan} == {}
         # (GH 6444, sorting of nans, is no longer an issue)
         index1 = Index([1, np.nan, 2, 3])
 
         result = index1.symmetric_difference(index2, sort=sort)
-        if sort:
+        if sort is None:
             expected = expected.sort_values()
         tm.assert_index_equal(result, expected)
 
-    @pytest.mark.parametrize("sort", [True, False])
+    @pytest.mark.parametrize("sort", [None, False])
     def test_symmetric_difference_non_index(self, sort):
         index1 = Index([1, 2, 3, 4], name='index1')
         index2 = np.array([2, 3, 4, 5])
@@ -1093,7 +1198,7 @@ def test_symmetric_difference_non_index(self, sort):
         assert tm.equalContents(result, expected)
         assert result.name == 'new_name'
 
-    @pytest.mark.parametrize("sort", [True, False])
+    @pytest.mark.parametrize("sort", [None, False])
     def test_difference_type(self, sort):
         # GH 20040
         # If taking difference of a set and itself, it
@@ -1104,7 +1209,7 @@ def test_difference_type(self, sort):
             expected = index.drop(index)
             tm.assert_index_equal(result, expected)
 
-    @pytest.mark.parametrize("sort", [True, False])
+    @pytest.mark.parametrize("sort", [None, False])
     def test_intersection_difference(self, sort):
         # GH 20040
         # Test that the intersection of an index with an
@@ -1607,11 +1712,11 @@ def test_drop_tuple(self, values, to_drop):
 
         ('intersection', np.array([(1, 'A'), (1, 'B'), (2, 'A'), (2, 'B')],
                                   dtype=[('num', int), ('let', 'a1')]),
-         True),
+         None),
 
         ('union', np.array([(1, 'A'), (1, 'B'), (1, 'C'), (2, 'A'), (2, 'B'),
                             (2, 'C')], dtype=[('num', int), ('let', 'a1')]),
-         True)
+         None)
     ])
     def test_tuple_union_bug(self, method, expected, sort):
         index1 = Index(np.array([(1, 'A'), (2, 'A'), (1, 'B'), (2, 'B')],
@@ -2259,20 +2364,20 @@ def test_unique_na(self):
         result = idx.unique()
         tm.assert_index_equal(result, expected)
 
-    @pytest.mark.parametrize("sort", [True, False])
+    @pytest.mark.parametrize("sort", [None, False])
     def test_intersection_base(self, sort):
         # (same results for py2 and py3 but sortedness not tested elsewhere)
         index = self.create_index()
         first = index[:5]
         second = index[:3]
 
-        expected = Index([0, 1, 'a']) if sort else Index([0, 'a', 1])
+        expected = Index([0, 1, 'a']) if sort is None else Index([0, 'a', 1])
         result = first.intersection(second, sort=sort)
         tm.assert_index_equal(result, expected)
 
     @pytest.mark.parametrize("klass", [
         np.array, Series, list])
-    @pytest.mark.parametrize("sort", [True, False])
+    @pytest.mark.parametrize("sort", [None, False])
     def test_intersection_different_type_base(self, klass, sort):
         # GH 10149
         index = self.create_index()
@@ -2282,7 +2387,7 @@ def test_intersection_different_type_base(self, klass, sort):
         result = first.intersection(klass(second.values), sort=sort)
         assert tm.equalContents(result, second)
 
-    @pytest.mark.parametrize("sort", [True, False])
+    @pytest.mark.parametrize("sort", [None, False])
     def test_difference_base(self, sort):
         # (same results for py2 and py3 but sortedness not tested elsewhere)
         index = self.create_index()
@@ -2291,7 +2396,7 @@ def test_difference_base(self, sort):
 
         result = first.difference(second, sort)
         expected = Index([0, 'a', 1])
-        if sort:
+        if sort is None:
             expected = Index(safe_sort(expected))
         tm.assert_index_equal(result, expected)
 
diff --git a/pandas/tests/indexes/test_range.py b/pandas/tests/indexes/test_range.py
index bbd1e0ccc19b1..96cf83d477376 100644
--- a/pandas/tests/indexes/test_range.py
+++ b/pandas/tests/indexes/test_range.py
@@ -503,7 +503,7 @@ def test_join_self(self):
             joined = self.index.join(self.index, how=kind)
             assert self.index is joined
 
-    @pytest.mark.parametrize("sort", [True, False])
+    @pytest.mark.parametrize("sort", [None, False])
     def test_intersection(self, sort):
         # intersect with Int64Index
         other = Index(np.arange(1, 6))
diff --git a/pandas/tests/indexes/timedeltas/test_timedelta.py b/pandas/tests/indexes/timedeltas/test_timedelta.py
index 547366ec79094..79210705103ab 100644
--- a/pandas/tests/indexes/timedeltas/test_timedelta.py
+++ b/pandas/tests/indexes/timedeltas/test_timedelta.py
@@ -51,7 +51,7 @@ def test_fillna_timedelta(self):
             [pd.Timedelta('1 day'), 'x', pd.Timedelta('3 day')], dtype=object)
         tm.assert_index_equal(idx.fillna('x'), exp)
 
-    @pytest.mark.parametrize("sort", [True, False])
+    @pytest.mark.parametrize("sort", [None, False])
     def test_difference_freq(self, sort):
         # GH14323: Difference of TimedeltaIndex should not preserve frequency
 
@@ -69,7 +69,7 @@ def test_difference_freq(self, sort):
         tm.assert_index_equal(idx_diff, expected)
         tm.assert_attr_equal('freq', idx_diff, expected)
 
-    @pytest.mark.parametrize("sort", [True, False])
+    @pytest.mark.parametrize("sort", [None, False])
     def test_difference_sort(self, sort):
 
         index = pd.TimedeltaIndex(["5 days", "3 days", "2 days", "4 days",
@@ -80,7 +80,7 @@ def test_difference_sort(self, sort):
 
         expected = TimedeltaIndex(["5 days", "0 days"], freq=None)
 
-        if sort:
+        if sort is None:
             expected = expected.sort_values()
 
         tm.assert_index_equal(idx_diff, expected)
@@ -90,7 +90,7 @@ def test_difference_sort(self, sort):
         idx_diff = index.difference(other, sort)
         expected = TimedeltaIndex(["1 days", "0 days"], freq=None)
 
-        if sort:
+        if sort is None:
             expected = expected.sort_values()
 
         tm.assert_index_equal(idx_diff, expected)