Chris P. answered 08/28/19
Experienced Teacher & Graphics Pro Keeps it Clear & Concise
Seems like a reasonable thing to want to do, but no, Illustrator CC currently can’t do that.
It's possible, though:
1) InDesign has a Pathfinder button that will equalize the handle lengths on a selected smooth point, which would achieve the same thing—symmetry on both sides.
2) Astute Graphics™ makes plugins for Illustrator that can do this and lots of other similar path-editing tricks. They have a demo available.