Is there a Möbius torus?

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/:

In the course of experimenting with 3-D models, I developed a method for twisting a toroid of arbitrary cross section. I called these forms Möbioids. Of course, if the cross-section is circular the result will be indistinguishable. What I did was to consider cylinders of non-circular cross-section. However, then the twist angles are quantized. Depending on the particular cross-section and number of twists, you can get forms with one or more surfaces. The figures below show an astroid cross-section with one twist of $\pi/4$ (left) and a pentacuspid cross-section with six twists of $2\pi/5$ (right). Each has a single surface. Again, not all twists lead to a single surface. You can find more images and some animations at A New Twist on Möbius¹.

¹Archived version in case the link above ever dies

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).