I've recently studied rolling polyhedra graphs. A polyhedron is rolled over its edges until it has reached all possible orientations. Ideally one more roll puts it back where it started. For example, the snub disphenoid has 72 orientations. If it is rolled through them all and ends back where it started with one more roll then there are 12 possible paths.
A bit too hard to simplify. But even the tetrahedron seems challenging. Do all the triangles need angles that are rational fractions of $\pi$, or does just the sum of angles around each vertex need to be a rational fraction of $\pi$? If the sum of angles around a vertex is not a rational fraction of $\pi$ then there are infinite orientations just rolling around that vertex.
A rolling non-regular tetrahedron has $n$ orientations. What values can $n$ be?