Suppose you have two bowling balls in outer space, connected
by a string, orbiting around the system CM like a barbell,
once per second. I claim that each ball is also spinning
around its own center, once per second. I further claim
that if the string snaps, each ball moves away along a
straight line, no longer orbiting the system CM ... but
still spinning!
Can I add another one? One of the first things I show students when we get to circular motion is an Interactive Physics simulation of a racecar going around a track. I just use it to show the direction of the velocity and acceleration vectors. The car is held to the circular path by a rope and after a couple of laps, I have the rope break. It was my intention to show that without the centripetal force, the car goes in a straight line. But the first time I ran it, I admit that I was surprised to see the car move off in a straight line, but also rotate -- it was actually a very realistic view of a car "spining out". But I wasn't expecting it and I thought "What makes the car start to rotate?". Then I realized that the car didn't START to rotate -- it was already "rotating" as it circled the track, just like the moon. In fact, observers in the center of the track would never see the dark side of the car. You can even time the rotations after the rope breaks and see that the period matches the period of the circular motion.
BTW if anyone wants a copy of the simulation, just let me know.