Problem 2.2.30 in Hatcher involves the homomorphisms $H_2(S^1\times S^1) \cong \mathbb{Z} \rightarrow H_2(S^1\times S^1) \cong \mathbb{Z}$ induced by
The map $S^1 \times S^1 \rightarrow S^1 \times S^1$ that is the identity on one factor and a reflection on the other.
and
The map $S^1 \times S^1 \rightarrow S^1 \times S^1$ that interchanges the two factors and then reflects one of the factors.
I strongly suspect that the induced homomorphisms are $-\mathbf{1}$ and $\mathbf{1}$, respectively. How would I prove this?