Orbiting Billiard Balls

This is a hard but hopefully interesting scenario.

Drawing out the table(s) and experimenting with different paths (angles) will generate a 'feel' for the problem and begin to suggest some features and underlying relationships.

Ideally the context is rich in possibilities for extension and fresh questions. For example are orbital paths possible where a rebound does not always take the ball on to the adjacent side but instead across to the opposite side, and if such paths exist, what is the connection between simple orbits and those where the cycle involves more than 4 rebounds?