This story begins in 1988 with the first examples of doubly periodic embedded minimal surfaces where the top and bottom ends are parallel and asymptotic to vertical half planes. They were found by Karcher and Meeks-Rosenberg in two independent papers and look like these:
I like it how confusingly similar they are. The main distinction is that that the one on the left has horizontal straight lines on it about which you can rotate the surface unto itself, while the one on the right has a reflectional plane of symmetry instead.
These surfaces actually come in a 3-parameter family; what you see above are the most symmetric cases. The translational periods are horizontal, and there are vertical straight lines. If you divide by the translations, you get a torus as a quotient surface. Remarkably, this 3-dimensional family is all there is for genus 1, by work of Pérez, Rodriguez, and Traizet from 2005. In particular this means that the two surfaces above can be deformed into each other through minimal surfaces. This is not too hard to see.
Things get more interesting already at genus 2: At 1992, Wei found a 1-parameter family of examples of genus 2.
A specimen is below to the left.
A variation of it was constructed by Rossman, Thayer and Wohlgemuth in 2000 (above to he right). Again they look dazzlingly similar. I suspect that they cannot even be deformed into each other through embedded minimal surfaces, but I have no idea how to prove this.
Even better, Connor has numerically found another genus 2 example that looks significantly different from the ones above. One of the holes has is not symmetrically positioned anymore, and there is no way to get it back there…
Showing the existence of these surfaces would be good, and even better would be to find a way to distinguish it from the others.