Klein bottle as the union of two Mobius bands

[math8]

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–Vietoris_sequence#Klein_bottle

AnB is also homotopic equivalent to a circle, I would think that since the intersection is the disjoint union of two Mobius bands, it is h.e. to the disjoin union of 2 circles, hence $$H_n(AnB)$$ should be $$Z \oplus Z$$ .

Am I thinking this all wrong?

 

[hunt_mat]

It's been a long time since I looked at this stuff but I think that it has something to do with wrapping around twice.

Mat