One way I've found that works is to use the scale x function in the transform behavior. For instance, I did some twins scenes by dragging in two instances of the same puppet, scale x -100 on one of them, and animate each one separately. Using the *.puppet feature they can even be rigged differently in the puppet panel.
Thank you for your help!
Heads up on using this method. If you have eye-tracking, it'll be backwards whether you're using the camera to track, the keyboard, or the mouse. So, be aware you'll have to do all the eye tracking in the "wrong" direction, either before or after you mirror the character.
Also, if you're character has any identifying features on one side (sword on his left hip or pocket-watch on his breast pocket) that'll be wrong, too.
One of the major things I want to propose to Adobe is a method to mirror rigs. Currently, I create all the layers in PS, then duplicate and mirror them, then I have a script this renames the layers from "Right" to "Left" and "Left" to "Right". That way, the left arm becomes the right arm (or eye or leg or whatever) instead of the left arm being on the right side of the character after the mirror process.
But, even then, you still have to manually re-rig both sides. Seems to make sense to have a more automagic function for that.