Is there a Möbius torus?

Question

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.

Answer 1

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 http://im-possible.info/english/articles/klein-bottle/:

Answer 2

You twist it by $\pi$ and you get that "Möbius torus" Möbioid.

Twist it by $\frac{2\pi}3$ and you get a nice impossible triangle

Penrose triangle https://en.wikipedia.org/wiki/Penrose_triangle

I just noticed that it regularly tiles the torus with three pairs of same colored toric (concave) rectangular hexagons, all those six faces connected exactly like a cube. Cubic contentedness (although different topology) but with hexagons, instead of squares...

You twist it by $e$, or any other irrational number, not necessarily transcendental, and any underlying astroid, rectangular other cross section will get smoothed out into a blurry thick-toroidal Möbioid ("surface" Hausdorff dimension of three, instead of normal two).