I made use of this astonishing (to me) fact a couple of years ago, in an article I co-authored on hypothetical physical theories [0].
We considered theories where the "Bloch sphere" [1] of a primitive system took the form of a regular convex polytope, and showed that this led to highly restricted multi-system dynamics. We used the fact that every facet was diametrically opposite to another facet, except in the case of odd-sided polygons, for which we had to drive the same thing in a different way. It wasn't a huge or surprising result, but it was a lot of fun to prove something that applied to those polytopes in all dimensions.
We considered theories where the "Bloch sphere" [1] of a primitive system took the form of a regular convex polytope, and showed that this led to highly restricted multi-system dynamics. We used the fact that every facet was diametrically opposite to another facet, except in the case of odd-sided polygons, for which we had to drive the same thing in a different way. It wasn't a huge or surprising result, but it was a lot of fun to prove something that applied to those polytopes in all dimensions.
[0] https://arxiv.org/abs/1508.03491
[1] https://en.wikipedia.org/wiki/Bloch_sphere