This animation made for the book http://www.motionmountain.net shows the Dirac belt trick for a 1/2 particle (the square object), behaving as a fermion by returning to the same state after a 4 pi rotation (but not after 2 pi). This is shown here with two attached belts, but the trick works with any number of belts!
When the particle is rotated 720 degrees, that produces a twist in the belts that might look as representing a different state. However the animation shows that this can be untangled without crossings by using the belt trick for both belts at the same time. This illustrates that the state after a 720 degree rotation is equivalent to the original.
This serves as a preparation for the next video illustrating the spin-statistics theorem for spin 1/2 particles:
More examples: http://www.cs.indiana.edu/~hansona/quatvis/Belt-Trick/index.html