Let $\Sigma$ be your surface.<br>
I'll construct a new surface $\tilde \Sigma$ as follows.
Take $\Sigma$ apart into individual triangles, and reglue them with a twist at each edge (the adjacency graphs for the faces of $\Sigma$, and for the faces of $\tilde\Sigma$ are the same). Note that the surfaces $\Sigma$ and $\tilde\Sigma$ could have different genus, and that $\tilde\Sigma$ could be non-orientable.

Your data is equivalent to having a piecewise linear map (not necessarily an embedding) of the new surface $\tilde\Sigma$ into $\mathbb R^n$, so that the vertices of the triangulation map to $\mathbb Z^n$.<br> That map is well defined up to an overall translation.