[iqzzgvod] How is an icosahedron like an even permutation?

Create a geometric demonstration of how the rotations of an icosahedron or dodecahedron align with the even permutations of 5 objects, in particular the group action.

My guess is there is no good demonstration: that the two groups are isomorphic is just coincidence, vaguely an instance of the Strong Law of Small Numbers: there are not enough small finite groups to meet the demands made of them.  The icosahedral group had to factor into some collection of small finite simple groups, and A5 is where it just happened to land.

Why is a raven like a writing desk?  Lewis Carroll was a mathematician.