By torus grid I mean a graph with $n \times m$ vertices $v_{ij}$ and edges $\{(v_{i,j}, v_{(i+1) \mod n,j})\} \cup \{(v_{ij},v_{i,(j+1) \mod m})\}$.
By automorphism here I mean a permutation $\pi$ over the set of the graph's edges such that there is a permutation $\psi$ over the set of the vertices such that $\pi(u,v) = (\psi(u), \psi(v))$