It looks like this never got a solution, so I'll just make some comments about this part of the problem here.
In particular, the biggest obstruction to getting a finite classification that doesn't lump too many things together is as follows: start with the standard "Latin cross" unfolding, which was the base of the one above. Note, in particular, the middle four squares forming a rectangle whose top is connected to its bottom.
That rectangle can be transformed by shearing such that the left and right sides are congruent to the original and connect in the same way, but the middle column instead wraps around like a cardboard tube. It's possible to make this wrap around an arbitrarily large number of times, and this number is robust to topological deformations like the one described above.
I do have a more full answer that should describe all those using combinatorial and geometric data, but it deserves a proper write-up instead of just a Substack comment.
I think the submission form is still open if you want to submit a larger writeup via a link. I'd be happy to share the writeup on the page. I didn't end up posting a solution here since the question was very open ended and I didn't get any submissions when I first posted it.
It looks like this never got a solution, so I'll just make some comments about this part of the problem here.
In particular, the biggest obstruction to getting a finite classification that doesn't lump too many things together is as follows: start with the standard "Latin cross" unfolding, which was the base of the one above. Note, in particular, the middle four squares forming a rectangle whose top is connected to its bottom.
That rectangle can be transformed by shearing such that the left and right sides are congruent to the original and connect in the same way, but the middle column instead wraps around like a cardboard tube. It's possible to make this wrap around an arbitrarily large number of times, and this number is robust to topological deformations like the one described above.
I do have a more full answer that should describe all those using combinatorial and geometric data, but it deserves a proper write-up instead of just a Substack comment.
I think the submission form is still open if you want to submit a larger writeup via a link. I'd be happy to share the writeup on the page. I didn't end up posting a solution here since the question was very open ended and I didn't get any submissions when I first posted it.