I'm aware that when you glue the ends of a rectangle together normally, you get a torus. I'm also aware that when you glue the ends of a rectangle together while introducing a half-twist to one of the sides, you get a Klein Bottle. The way that each of these things gets glued makes sense to me, and the homeomorphism is pretty clear.
But what happens when you try and introduce a half twist to both sides? Is it even possible to do this? I've been fiddling around with cut-and-paste methods to work out what this might be and I'm coming out inconclusive.
It's the real projective plane. Its Wikipedia page provides diagrams much the same as yours.