In theory, the Klein bottle has only one side, but in our world, this cannot be so. We have to have an inside and outside, and it is tricky to hide this. OpenGL has a nice feature that lets you illuminate both inside and outside equally, as in the illustration above. If you do not use this, it becomes clear where the inside and outside meet:
Here is a close up, moving just inside the neck of the bottle so that the surface is cut away.
The top half of the picture shows the brightly lit outside on the inside of the bottle, and the darkly light inside on the outside of the bottle. The bottom half has this reversed, as would normally be the case.