ER, from what I remember mentioned in astronomy magazines, it's the solar system that travels from one density wave to the other. To describe the phenomenon, the analogy with tafic jams is used. Every car being a sun, sometimes cars get slowed down in trafic jams and then re-accelerate (cruising speed) until they get again cought in a trafic jam. Now, does this actually happen riding backwards, I don't know. What does happen though, is that it seems the speed of our Sun around our galaxy is in accordance with its distance from its center.

When we look at a galaxy,there's a side emitting light towards the redshift (gettting away from us) and a side emitting light towards the blueshift (or less of a redshift) showing the stars traveling towards us.

The way galaxies rotate has to do with how they got formed (merging with other galaxies, colliding at an angle, which sends streams of stars in a direction,which determines how the rotation will change).