Flip the hats!
The hats can only be rotated or moved. But for some avatars, the hat may look better (at discretion of the hat and avatar owner) if the hat was flipped over. This would make hat wearing so much cooler and make our avatars look even cooler (no Winter pun intended).
For example, take my current hat (Like Clockwork) on my Double 0-Minion avatar. The gear is next to the eye, but my Minion would look more spy-like with the gear over the eye. Only possible with hat flipping.