Homework Problem

Here is a figure from some notes I am reading:

My understanding of it is that we have a stiff cube-shaped frame, say made out of wooden dowels or thick wire, fixed in space. In the center of the frame we have a ball, attached to the corners of the frame with 8 rubber bands.

The claim is that if we rotate the ball 360 degrees around the vertical axis, and hold it there, we would be unable to untangle the bands without moving the ball. However, if we rotate the ball 720 degrees around the vertical axis and hold it there, then we could get the bands back how they were when we started.

Wow. Can you see in your head how this works? I'm tempted to build one. It seems weird to me. At what degree of rotation does this untangling become possible? Supposedly it's more than 360 and less than or equal to 720 degrees. How about 719 degrees? How does this magically jump from "not possible" to "possible" as we continuously rotate the ball?