Today I learned about the figure-8 immersion of the Klein bottle. Instead of starting with a tube and then sticking the tube through the wall of the tube to get it to join up in the opposite direction, you start with a double tube whose cross-section has the shape of a figure 8. Then you bend the double-tube around into a circle and join the ends, but you give the double-tube a half-twist as you bend it, so that it turns into a single tube with a half twist.

The self-intersection is still a circle, but instead of being the place where the tube punches through the wall of the bottle, it's the set of crossing points of the figure-8s.

https://en.wikipedia.org/wiki/Klein_bottle#The_figure_8_immersion