Regular Hamiltonian embeddings of the complete bipartite graph $K_{n,n}$ in an orientable surface

An embedding $M$ of a graph $G$ is said to be regular if and only if for every two triples $(v_1,e_1,f_1)$ and $(v_2,e_2,f_2)$, where $e_i$ is an edge incident with the vertex $v_i$ and the face $f_i$, there exists an automorphism of $M$ which maps $v_1$ to $v_2$, $e_1$ to $e_2$ and $f_1$ to $f_2$. We show that for $n\not\equiv 0$ (mod 8) there is, up to isomorphism, precisely one regular Hamiltonian embedding of $K_{n,n}$ in an orientable surface, and that for $n\equiv 0$ (mod 8) there are precisely two such embeddings. We give explicit constructions for these embeddings as lifts of spherical embeddings of dipoles