Is there a gadget that reduces generalized geography to undirected graphs?
Undirected (Vertex) Geography is in P. In particular, the game on graph $G$ with starting vertex $v$ is a win for player 1 if and only if every maximum matching of $G$ uses the vertex $v$. This can be checked in polynomial time.

The above is Theorem 1.1 from the paper "Undirected Edge Geography", by Fraenkel, Scheinerman and Ullman, Theoretical Computer Science 1993.