This is one of two families of genus 5 surfaces that have the same behavior along the boundary of the box: In one boundary component, the Gauss map points vertically up at two corners, while in the other component it changes between up and down.
The difference is in the behavior of the Gauß map along a horizontal symmetry curve: Here it rotates by 90º. All this is encoded in the divisor of the square of the Gauss map below, with the stipulation that a+b+c=d+.5.
There are no special subcases with more symmetries and easier period problem. Some limits involve doubly periodic RTW surfaces and Wei surfaces and require catenoidal stitches, which is beyond the current state of the arts.
Fortunately, the surfaces can be obtained via Traizet regeneration from a nodal limit: