A fold means two different regions of topology get projected across each other.

It's a problem for the simplest of reasons, information is lost. You cannot reconstruct the original topology.

In terms of the model, it now can't distinguish between what were completely different regions.

From the Klein bottle perspective, a 4D shape gets projected into a 3D shape. On most of the bottle, there is still a 1 to 1 topological mapping from 3D to 4D versions.

But where two surfaces now intersect, there is now no way to distinguish between previously unrelated information. The model won’t be able to anything sensible with that.

TLDR; We don't like folding.