Quantum spin is described with spinors. If you want geometry, one may observe the celebrated Dirac belt trick. It takes 720 degrees of rotation to return to its original configuration.

But I have long thought that things had to be simpler than that. Electrons and so forth are extremely simple objects. The Dirac belt trick topology seems too complex for them.

Here's a simpler way to look at it. Consider that quantum spin can't be observed directly. It can (I think) only be observed as the result of of the collision of two spinning objects. Let's go with that. It is also clearly an even-dimensional phenomenon. How about 2.

Consider two coins. Label each point on the perimeter with an angle (polar coordinates). Let's have theta for coin 1 and tau for coin 2. Now you want the coins osculating so that the point at which they make contact is such that theta=tau=C. To do this, coin two has to be upside down.

Now start rotating the two coins clockwise. As soon as this begins theta is no longer equal to tau until we have rotated each coin 180 degrees. Theta=tau=C+pi.

So each coin has rotated only 180 degrees. But all we observe is the phase between the two coins. This has cycled through 360 degrees.