As I understand it, galaxies don't rotate like rigid bodies: stars further from the centre take longer to go round than ones closer in, much like planets further from the sun take longer to complete an orbit.

If stars that start off inside an arm remain so, but the outer part of the arm goes round more slowly, the arm would get wound 'tighter' as time goes on, and this doesn't seem to happen.

I think what's really going on isn't yet well understood, but it's likely to be that the arms are something more like a standing pattern, and stars do indeed move in and out of the arms as time goes on (something like the way a traffic jam can move more slowly than the individual cars that it's made of).