"I watched your icon go around a few times until it dawned on me, "Oh, it found itself and disappeared. Hey, that's an identity." Then I counted how many transformations it made (eight) before it disappears.
My favorite algebra book is Topics in Algebra by I.N. Herstein, and one of the first examples he uses to introduce group theory is the group of rotations that maps a square onto a square. That idea has always appealed as very simple and intuitive, but very significant and powerful. Which I think is why it caught my eye.
I wanted to say that it's a quaternion group, but upon reflection I don't think that's so; it appears to be commutative, so it's probably just isomorphic to Z/8 under addition, with eighth-turns odd and quarter-turns even.
I've got a feeling you were probably deliberate about the design; what did you have in mind?"