Torsion in the Twists (Belt/Plate Trick)

This is a follow-up post to the statement at the end of Everyday I'm $\mathbb{RP}^{n+1}$, that $\pi_1\left(\mathbb{RP}^3\right) \cong \mathbb{Z}_2$

$\mathbb{Z}_2$...  Isn't that the finite simple group of order two?

It was only a matter of time before I found an excuse to post this...

Anyway, have some links:

A quick video of the corresponding plate trick* 

Brief discussions here and here of the belt/plate trick, courtesy Wikipedia.
Some nice demonstrations (and also discussions) of the tricks, courtesy Bob Palais.
A quick video of the much-related quaternion handshake**

*In the plate trick, a path in $SO(3)$ corresponding to a $720^\circ$ rotation is traced out over time by your hand, and the homotopy to a $0^\circ$ rotation is traced out 'spatially' as you move up/down your arm.  These roles are reversed in the belt trick, where the path is traced out spatially and the homotopy occurs over time.

**$\mathbb{RP}^3$ is a quotient of $S^3 \subset \mathbb{R}^4$.  The quaternions are isomorphic to $\mathbb{R}^4$ as an $\mathbb{R}$-vector space.
So it shouldn't be too surprising that $SO(3)$ is a quotient of the unit quaternions.