What determines which character I play in Portal 2 co-op?

At the moment it is not really possible choose the robot. The player that hosts the game is automatically assigned Atlas, the friend that joins the game has P-Body instead.

Here's what I saw: I hosted the first game with my friend. I was assigned Atlas, she was P-Body. The next time, she invited me. Again, I was Atlas and she was P-Body. The third time, I invited her. Same result: I had Atlas, she had P-Body.

My guess is that, the first time you play with someone, the inviter is given Atlas and the invitee is given P-Body. Upon subsequent plays with the same person, each keeps the same robot no matter who initiates.