@EricRogstad I believe that Brendan was saying that a Rubik's cube is a Monoid.
Even thought the rotations change the way it looks and it's orientation, the Rubik's cube is still a Rubik's cube. In the same way that a monoid where the arrows are the natural numbers and addition is the way you compose them doesn't change the underlying object when you compose 2 and 3.
Another way to think of it, assume that the Rubik's cube is only changed while you are observing one of the arrows. Once you stop looking at the arrow it goes back to its "resting state".
My question would be, in a Rubik's cube, if it is indeed a monoid, what is the identity arrow?