Removing intersection on pipes

Question

I was modeling a Klein Bottle and end up in this type of problem after doing Solidify:

I managed to remove the intersections on the Klein duplicating the surfaces and doing Boolean operations with a lot of manual deleting afterwards.

Now I am trying to find a better approach for this problem. I think the easiest solution would be to duplicate both pipes and separate them, after that doing a Intersect Boolean and a Difference in sequence; but it is not removed completely. How it is possible to fix this?

Edit: This pipe problem is a simplification of the Klein. It not about the steps to create pipes, but to fix the mesh obtained by complex XYZ Math Surface:

Answer 1

Here is my approach from long ago, although I would do a better job with the angles now:

• Create an UV sphere.
• Delete the top and bottom vertices.
• Extrude a 'column' from the top into the sphere.
• Subdivide one of the faces in the sphere enough times to create a circle.
• Delete all of the faces inside that circle.
• Bridge edge loops between the previous column and the new circle.
• Bring a column around from the bottom to the same circle, by extruding and rotating.
• Add a subsurf

If I were to do it now, I'd use a sphere with fewer slices, I would spin the turns in the columns to make them smoother and I would make the circle from several faces.

Alternatively, one could just enable Extra Objects, add an X, Y, Z Function Surface, and choose the Klein preset:

giving

EDIT: If I wanted to 3D print the original, I would modify it so that it was symmetric on the XZ plane, cut it in half and print the two halves. This way you wouldn't have any extra faces to delete, although depending on the printer you would probably need some interior supports you would need to remove after printing. It would look something like this with the subsurf in place: