This surface enjoys a hard to spot additional orientation reversing symmetry: Reflect at a horizontal plane, and translate diagonally and horizontally by half-periods. This makes the period problem 3-dimensional.

As limits you can choose between two doubly periodic Scherk surfaces glued end-to end with double catenoidal stitches (for safety), or a pair of KMR surfaces, also glued end-to-end.

Resources

Mathematica Notebook

PoVRay Sources