In OEIS A003436, it is written that the number of inequivalent labeled Hamilton Cycles of an n-dimesnional Octahedron is the same as the number of Perfect Matchings in a the complement of the Cycle Graph on 2n vertices (i.e. a complete graph $K_{2n}$ without the cycle $C_{2n}$).

I can verify this relationship, but it seems very arbitrary to me. Unfortunatly, OEIS does not give further details.

Why are these number of these two properties (inequivalent Hamiltonian Cycles and Perfect Matchings, respectively) equivalent in these two different objects (n-dimensional Octahedron and Complement of Cycle Graph $C_{2n}$)?

And hint or reference to the literature is very much apprechiated.