Is a mirror-reversing path through the universe a possibility from a physics point of view?

[hurrycane]

Recently I watched a couple of fun YouTube videos on Klein bottles. In one of them

 

https://youtu.be/4XN0V4xHaoQ

 

the Mathologer, a math professor I think, talks about the mathematical possibility of a universe being shaped like a 3d counterpart of a 2d Klein bottle surface. That would mean that somebody living in such a universe could turn himself into his mirror image by travelling along certain closed paths. Is there anything in physics that would indicate that such mirror-reversing paths are or are not present in our universe?

 

[imatfaal]

We don't see right handed neutrinos nor left handed anti-neutrinos (presuming they are not majorana particles) - surely we would expect that some of the neutrinos we detect would have been flipped in their helicity/chirality by these strange paths