Imagine a ball in the middle of the room suspended in the air by six elastic cords. Call them ropes. Each rope is glued to the ball at one rope-end and to a wall, ceiling, or floor at the other rope-end. The ropes are all straight and untwisted.
Then the ball starts turning around on its vertical axis. The rope-ends attached to the ball’s surface go around with it while the rope-ends attached to the room remain stationary. The ropes become tangled and twisted.
The ball turns around exactly once and pauses. During this pause you can try to untangle the ropes but you won’t succeed.
Then the ball continues turning around, in the same direction, and stops after a second turn. Surprise, now you can untangle the ropes so they are all straight and untwisted, as they were at the beginning.
The manipulation can be done while the ball turns, then the ball can spin without pause forever yet the ropes never tangle and twist beyond a certain amount. You can do this with any number of ropes, six was just to be definite.
The program Antitwister shows movies of this motion. It can also show the ball spinning while attached to the room by an elastic web that completely encircles the ball.