Everyone knows that a 360° rotation brings an object back where it started.
But sometimes it doesn't.
Here's a demonstration you can do right now--and you should, because this trick is too strange to believe unless you do it yourself.
Take a small object. It doesn't much matter what it is--a business card, a pen, a water glass--just something you can keep vertical as you rotate it in the horizontal plane.
Hold out your right hand, palm up. Use your left hand to lower the object into the grasp of the fingers and thumb of the right hand. This hand and the object will move together from here on.
First rotate the object counterclockwise (looking down) by moving your elbow to your right and your hand to your left. Keep going as the object passes under your arm. To finish a full rotation, if you're joints are like mine, you'll have lift the object up to face level.
At this point, the object has completed a full rotation, and is oriented the way it started. But your contorted arm is telling you that not everything is the same!
Obviously one way to return to the starting point is to reverse the motion that got you here. But there's another way: keep going.
Your arm may object to the idea of becoming even more twisted, but bear with me. Holding the object over your head, keep rotating it in the same direction you were turning it before. This time, though, instead of your arm passing over it, your arm will pass under it.
If you did this right, when you completed the second full rotation, both the object and your arm were now back in their comfortable starting position. Cool, huh? Once you get good at it, you can do it with a partially-filled glass of water, which should convince you that you aren't flipping it over at some point in the motion.
The trickiest part of this trick is that it's not a trick.
This is a little-known property of three-dimensional space: although one full rotation leaves a disconnected object unchanged, two full rotations leave objects unchanged even if they are connected to the (non-rotating) rest of the world. It's not a property of your arm. You can connect the object to its surroundings with as many rubber bands as you like, and you will always be able to untangle them after two full rotations.
Is this just a strange factoid? Maybe. But consider that many elementary particles, notably protons, neutrons, and electrons, have this same property: it takes two full rotations to bring them back where they started. Not only that, but swapping any two of these particles leaves a clear quantum-mechanical signature. (I once saw the mathematical proof that the rotation and swapping properties are intimately connected in an advanced physics class. I was very proud to understand it. For a few hours.)
Not to get all new-agey, but these properties make a lot more sense if you envision electrons as embedded in the larger universe, rather than as independent particles floating freely in space.