Turing's Symmetries

Solution 1:

Imagine looking through the $P$ that's closer to you (colored in green below) and right at the $P$ that's on the far side (in red). Namely, you line of sight roughly aligns with the straight line formed by connecting the two $P$s.

Apologies for the choice of colors. I tried to make the green $P$ thicker to represent that it is closer to your eyes.

The red $P$ is an image of the green (thicker) $P$ via moving along the surface of cylinder (equivalent to rotating in space).

The upright stroke (the "vertical" bar when P is written this way) of one is not parallel to the other in the 3-dim Euclidean space.

^{I don't know how to produce using Mathjax the diagram with two $P$s. The all-powerful Detexify returns nothing.}