How to Setup Puppet with a Left and Right Hat Profile?

What I'm trying to do:

I'm using the Sticky Puppet and I want to put a hat on him that works in both left and right profile while walking.

Problem I'm having:

I'm dropping the hat on him inside the rig section on his head layer. This creates a copy of the hat in both left and right profiles with the hat pointing in one direction. So when I turn to walk the other way, the hat is facing the wrong direction. How do I make the hat turn with him?

When I try to edit the left profile hat, it effects the right profile hat and vice versa. How can I make the hats work independently of each other?

UPDATE:

I figured out how to do this! I dropped the hat on both profiles above the head then I dragged the Origiin down to the head until I got the green highlight and now they both show up correctly.

-Shaye