This surface has an orientation reversing symmetry that reflects at a horizontal plane, then translates by one half-period. The period problem becomes 3-dimensional. One of the limits is easy to guess: Two doubly periodic Scherk surfaces with carefully spaced double stitches. These examples indicate that a precise description of this gluing procedure will be tricky.
The other limit is simpler and explains what happens to the little catenoidal necks when pushed towards each other. We obtain 10-ended singly periodic Scherk surfaces. The arrangement of the 10 ends is amusing.
Below is an animation of the entire family: