Many moons ago I learned that electrons needed to rotate 720 degrees before coming back to their starting point. Among other implications, this means that if one constructs an apparatus to measure the spin of electrons emitted from a sample, then rotates the apparatus by 360 degrees (leaving the sample un-rotated), then the readings will change. A further rotation, for a total of 720 degrees, would bring the readings back to the original setup.
More recently I read Grahm Farmelo's biography of the famous physicist Paul Dirac. Paul graced us with a simple geometrical explanation for this kind of 720 degree rotation symmetry that applies even to macro objects. It's known as the Dirac String Trick.
There are many good animations of Dirac's String Trick on the web, but no physical models I could find, so I made one and posted the video here:
The idea is to suspend a ball on rubber bands spanning the x,y,z Cartesian coordinates. The rubber bands can stretch, but must remain connected. The video shows Dirac's trick of how the bands may be unwound after the ball is rotated 720 degrees.