Hi pattern guys!

I can't get a straight answer to this question, perhaps you could help me!

I am thinking (and searching on the Net) about differential ailerons, and I can't figure out the REAL reason for they to exist in a pattern plane. All people say that they act in order to contrast inverse yaw. But, what I can't understand is this: if we have a symmetrical airfoil (in a pattern aircraft) and we are doing a vertical roll, what is the wing that is producing more lift (and hence more drag)?
Morover, let's suppose we hare differentiating ailerons in order to contrast inverse yaw. Well, when we are in inverted flight and we do a roll, they act in the opposite direction! The aileron goes down much more, obviosuly. Needless to say that differential ailerons do help when doing rolls, and not the opposite!

These argumentations suggest to me that inverse yaw is not the real motivation, at least referring to pattern aircraft, where aileron differentials are still necessary!

I hope you will have the time to help me with this little mistery (for me)!!!

Thank you in advice and good flying!