Thanks for sharing. I'm very familiar with the basic mechanics of quaternion rotation, and I've been interested in a deeper understanding of this double-cover concept, but I just don't get it. I've seen the belt trick and it feels more like an illusion than an illustration of some deep truth.

I like how you've connected it to spin, but I still don't understand how that is a real physical property rather than a mathematical artifact.

I don't quite grasp the significance of your "different look". Can you suggest any other reading?