If you want to animate the geometry itself, you can think of your tube as a stretched-out torus. The torus in turn is made of a line of cylinders, curved into a circle. You can make something like your shape from one cylinder with array, curve, and lattice modifiers added to it in turn, like this crude example:
Select the highly modified cylinder object and rotate on its (local) Z axis, and you’ll get the motion you want.
Notes on construction:
Make sure the 3D cursor is always at the world origin for all these procedures.
Add the original cylinder in Top view, so the global Z axis is parallel to its long axis. Remove the end faces.
The Array modifier should be set to offset along Z.
Add a Bezier circle and use it as a Curve modifier on the arrayed cylinder. Adjust the array count so the cylinder segments go all the way around to form a torus.
Add a Lattice and use it as a Lattice modifier on the cylinder. In Edit mode, scale the Lattice on its Z axis to stretch the torus into an open cylinder.
You can UV unwrap the original cylinder to add a texture, but the texture would be highly distorted by all the modifications. I guess you could start with an already distorted texture; I haven’t tried it myself yet.
You could also dispense with the array modifier and start with an elongated, subdivided cylinder. It might be easier to texture, although you’d still have the stretching problems.
Here is a .blend file showing the rudiments: WRING torus.blend (488 KB)
I got interested in this inside-out/wringing effect while thinking about the end of Terminator 2, when the T-1000 is writhing around in the molten steel. The character turns inside out through its mouth while morphing into shapes it mimicked earlier. I’m still trying to get my head around how to simulate something like that in Blender.