Euler class is odd under orientation, thus its integral over a manifold will be even.

Question

I learned a statement from others:

"Euler class is odd under orientation, thus its integral over a manifold $M$ is even."

I cannot fully appreciate it, can someone show this explicitly?

The even/odd here I suppose it means the orientation.

Answer 1

This just means that the Euler characteristic of the manifold is independent of the orientation: $$\chi(-M)=\chi(M)$$