In what follows $T$ is the torus and $K$ is the Klein bottle.
I am just looking at Hatcher's example calculation using cellular homology of $T \times S^1$ and $K \times S^1$. Hatcher's homology notes are available here (see page 142 for question, and the commutative diagram and corresponding notes on page 141 for the definition of $\Delta_{\alpha \beta}$)
Hatcher states (in regards to $T \times S^1$):
Each $\Delta_{\alpha \beta}$ maps the interiors of two opposite faces of the cube homeomorphically onto the complement of a point in the target $S^2$ and sends the remaining four faces to this point. Computing local degrees at the center points of the two opposite faces, we see that the local degree is 1 at one of these points and −1 at the other, since the restrictions of $\Delta_{\alpha \beta}$ to these two faces differ by a reflection of the boundary of the cube across the plane midway between them, and a reflection has degree −1.
At first glance, I would have said that the faces have degree two. Are they a reflection because the faces are oriented in the opposite direction?