In Pose mode, select the bones you want to flip, then go to Pose > Flip IK Limits.
This tool is currently only designed with basic IK in mind. This does not work for things like position or rotation remap constraints.
I completed my rig for the right-side of an armature, but when I ran Symmetrize in Edit Mode, the bones on the armature's right (left in picture) had the wrong IK limits! This means the arm will move awkwardly and backwards when we move our IK goal.
To fix this, I selected all the bones on the right half of the armature (our left when facing it), then ran Pose > Flip IK Limits on the Z axis.
Now our IK limits are the same!
