Center of the mapping class group of the torus ($Z(\mathrm{Mod}(T^2))=\{1,\iota \}$)

The center of $\mathrm{Mod}(T^2)$ has order two. A central class commutes with twists about two curves intersecting once, hence preserves both curve classes. The Alexander method reduces it to the identity or the elliptic involution $\iota :x\mapsto -x$, and both are central.