Klein Bottle as the connected sum of two projective planes

Question

Can someone explain intuitively why the Klein bottle is diffeomorphic to the connected sum of two projective planes? I can do this using origamis/fundamental graphs )w/e they are called. is it possible to do it with another method?

Answer 1

I have a preferred way to explain this which involves the visualization of Klein bottle and its decomposition into two mobius strips (note that a projective plane minus a disk is a mobius band, hence the equivalence) as rectangles with certain sides identified.

You can find here two illustrations, the latter being the one I mentioned above. Actually the whole document is about answering your question in detail.