Biembeddings of Latin squares obtained from a voltage construction

We investigate a voltage construction for face $2$-colourable triangulations by $K_{n,n,n}$ from the viewpoint of the underlying Latin squares. We prove that if the vertices are relabelled so that one of the Latin squares is exactly the Cayley table $C_n$ of the group $\mathbb Z_n$, then the other square can be obtained from $C_n$ by a cyclic permutation of row, column or entry identifiers, and we identify these cyclic permutations. As an application, we improve the previously known lower bound for the number of nonisomorphic triangulations by $K_{n,n,n}$ obtained from the voltage construction in the case when $n$ is a prime number.