Degree of smooth map of manifolds depends on orientation choice?

Question

I'm a little to confused as to why it appears that the degree of a smooth map $f: M \to N$ between smooth manifolds appears to only be defined up to sign - I'm not sure where my mistake is.

By definition $\textrm{deg}(f)$ is the unique integer such that:

$$\int_M f^*\alpha = \textrm{deg}(f) \int_N \alpha$$

Where $\alpha$ is any n-form on $N$. Now it seems that if I were to pick different orientation on $N$ then the integral on the right hand side would switch sign?

Answer 1

I think you're onto something that is fully intended.

edit: here's a quote I think could be relevant

source on google books