Thanks, that's exactly the kind of comments I was looking for. That was exactly what I thought in my head, but it came out very distorted and ugly on paper. I'll update the solution later. Do you want credit?
Credit for this is optional. Another way to see obtain a partition of G is to observe that the relation x ~ y if and only if x = y or x = y^{-1} is an equivalence relation on G. The equivalence classes induce the partition above. The point is that any two equivalence classes {x, x^{-1}},{y,y^{-1}} are either disjoint or equal, and their union is G. But this is obvious (the word 'obvious' means "I thought of it.")
