Hamiltonian cycles in subdivided doubles

The subdivided double construction on 4-regular graphs was used by Potočnik and Wilson to explore semi-symmetric graphs. It preserves the property of being semi-symmetric, and can be used to construct every semi-symmetric 4-regular graph that contains a pair of twin vertices. We show that (regardless of symmetry) subdivided doubles have another curious property: each of their many Hamiltonian cycles is complementary to another Hamiltonian cycle.