In the context of topological quantum field theories, I am interested in the mapping class group of a torus. Here I can consider the torus as a square with identified edges and also decorated with directed curves (see below).

The mapping class group is $\mathsf{SL}(2,\Bbb Z)$ and is generated by $S$ and $T$, which correspond to a 90deg rotation and a Dehn twist respectively. I take the Dehn twist to be along the vertical (meridian). These generators should satisfy $S^2=C$, and $(ST)^3=S^2$, where $C$ is conjugation corresponding to flipping both directions of the torus. The first is easy to see

However, I cannot get the second property, here is my working

Note in this second case I am not mapping back to the original square using the periodicity to make it clear what transformations I am doing.

$(ST)^3=\Bbb 1$ is true for the modular group $\mathsf{PSL}(2,\Bbb Z)$ but then we should also have $S^2=\Bbb 1$.

Can anyone point out where I am going wrong?