his animation made for the book http://www.motionmountain.net shows how two spin 1/2 particles (represented by the blue and red square objects) behave as fermions under exchange. The usual belt trick for one particle, shown the previous animation vimeo.com/62228139, illustrates that such particles come back to their original state after a rotation by 4 pi, and thus behave as spin 1/2 particles. Together, the two animations thus visualize the spin-statistics theorem for spin 1/2 particles.
When the position of the two particles is exchanged twice (but not once) this results in tangled belts. However this animation shows how this can be untangled, making the final state equivalent to the initial
This extends the applet shown at http://www.gregegan.net/APPLETS/21/21.html to a situation with two belt buckles and four belts