Is there a Möbius torus? Does the concept of a _Möbius torus_ make sense: taking a cylinder (instead of a rectangle as in the case of the Möbius strip) and twisting it before joining its ends? Or will the resulting twisted torus be indistinguishable from the normal torus in any relevant respect? _[This equivalent to the well-known Möbius **strip** should be called Möbius **cylinder** but it would have so much in common with a torus that I preferred to call it a Möbius **torus**.]_ Embedded in Euclidean space the twisted and untwisted torus "look" the same - opposed to Möbius strip and cylinder -, the difference would be only in their intrinsic properties. But can there be such differences? And how do I specify them? PS: I posted a follow-up question here.

As remarked above: What you get is the Klein bottle. Put differently: The result is what you get when you take two Möbius strips (which both have one boundary) and glue both boundaries together (which does not work when embedded in 3d space but works in theory). See this image from < !enter image description here