On biembedding an idempotent Latin square with its transpose

Let L be an idempotent Latin square of side n, thought of as a set of ordered triples (i, j, k) where L(i, j) = k. Let I be the set of triples (i, i, i). We consider the problem of biembedding the triples of L - I with the triples of L' - I, where L' is the transpose of L, in an orientable surface. We construct such embeddings for all doubly even values of n.