Klein bottle as the union of two Mobius bands

I am trying to calculate the homology groups of the Klein bottle. I want to use the Mayer-Vietoris sequence with the Klein bottle decomposed as the union of two Mobius bands (A and B which are homotopic equivalent to circles), now AUB is the Klein bottle, but I don't understand how according to

http://en.wikipedia.org/wiki/Mayer%E...e#Klein_bottle

AnB is also homotopic equivalent to a circle, I would think that the intersection is the disjoint union of two Mobius bands, so it is homotopic equivalent to the disjoint union of 2 circles, hence $\displaystyle H_n(A n B)$ should be $\displaystyle Z \oplus Z$ . But according to what's written in wikipedia, $\displaystyle H_n(A n B)$ is just $\displaystyle Z $ . Why?

Am I thinking this all wrong?