The paper On the Planar Split Thickness of Graphs shows how non-planar graphs can be split to make planar graphs. For example, they offer a split $K_{6,10}$.
I would instead like to make split toroidal graphs. The Shrikhande graph (below), $K_7$, and the Möbius–Kantor graph are some toroidal graphs.
It seems that $K_{13}$ (2), $K_{19}$ (3), $K_{25}$ (4), $K_{31}$ (5), $K_{37}$ (6), $K_{43}$ (7), and $K_{49}$ (8) could all be realized as split toroidal graphs. In particular, can a $10 \times 10 $ grid with 1-25 each occurring 4 times be made so that all orthogonal or right diagonal pairings are distinct?
Here is a non-solution for $K_{25}$ as a toroidal 4-split with \ diagonals. Only 282 of the 300 edges are present. Above the grid are graphs made from doubled edges and missing edges.
