Took some finicky fiddling but finally got a hand-drawn Klein bottle with all the normals going the right way, and was able to import it into TopMod and play with it.

The 'simplest' variant has not been subdivided in any way... it's a good starting point if you want to adjust the overall shape.

The 'skin' variant has been subdivided using the Doo Sabin remeshing option in TopMod, giving it a rounded, smooth look. It might make a very infuriating teapot, I suppose.

The 'bone' variant is the same as the 'skin', but with more visibility inside.

To get a good idea of what is going on inside, some of the pictures of the wireframe views of these Things are your best bet.

So this answers the question: Yes, we can make a Klein bottle.

EDIT: Added an anaglyph image, since this is a confusing shape. Getcher red'n'blue glasses, step right up!

I made no effort to make this printable without supports, but the shape is watertight and no thinner than about a millimeter, so with supports, it should print just fine.

If you want to make your own derivative of this Thing, download the .obj file on this page and open it in TopMod or in Blender.

You can get TopMod here: topmod3d.org/

And Blender here: blender.org/