Graph embeddings on an orientable and nonorientable surface

Question

Let $G$ be a graph of orientable genus $M$ and nonorientable genus $N$ where $M\ne 0\ne N$. If we embed $G$ into an orientable surface of genus $M$ and a nonorientable surface of genus $N$ such that neither embedding contain an edge crossing, will the faces of the two embeddings necessarily be nonisomorphic? As in, will the edges of the two embeddings determine different faces?

Answer 1

$K_7$ which has orientable genus $1$ and nonorientable genus $3$ is a counterexample.