elements of $SL(2,\mathbb{Z})$ and their negative multiples acting with the mapping class group of the two-torus $T^2$

Question

Given some element A of $SL(2,\mathbb{Z})$ which is associated with a particular orientation preserving homeomorphism up to isotopy (by the isomorphism $\phi$), what orientation preserving homeomorphisms up to isotopy will the negative multiple of A in $SL(2,\mathbb{Z})$, that is $-1* A$ correspond to with respect to A? For instance, when considering the generators of the fundamental group of the torus, do both A and $-1*A$ "act" on these two generators similarly? Thank you.