Does the Universe SPIN once every 500 billion years?!

The question is whether or not our motion toward the great attractor is the universe circling the drain. We know we are heading there, but cannot see where places where the light has not yet reached us are doing the same thing. The question should be, at what point in the aging of the universe did we start the journey back to the initial point? The  distance at which observations shift from expansion to travel to the G.A. is the clue to demonstrating whether the question is reasonable.