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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.

enter image description here

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    $\begingroup$ The real projective plane? $\endgroup$ Aug 16 '18 at 15:42
  • $\begingroup$ Aw, it is? That's so boring. Thanks, that ought to be enough for me to look more into it. $\endgroup$ Aug 16 '18 at 15:44
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    $\begingroup$ Why is that boring? $\endgroup$
    – Randall
    Aug 16 '18 at 15:45
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    $\begingroup$ Note that unlike the torus and Klein bottle cases, all four corners of the rectangle do not come together by this gluing. $\endgroup$ Aug 16 '18 at 15:47
  • $\begingroup$ Oh, wait a minute, the real projective plane might be something totally different from what I thought it was. At first glance, I thought it was just $\mathbb{R}^2$ projected onto a surface. $\endgroup$ Aug 16 '18 at 15:49
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It's the real projective plane. Its Wikipedia page provides diagrams much the same as yours.

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