These surfaces have a similar description as the surfaces of type (++|+-)a in the sense that the Gauß map is pointing up or down in the same pattern at the vertices of a reflectional fundamental piece.
Here, however, the Gauss map rotates by 0º or 180º along the horizontal symmetry curves. Hence, while the divisor of the squared Gauss map also follows the same pattern, its spin structure is constrained differently by d=a+b+c.
There is a subcase with rotational symmetry of order 4 about a horizontal axis. This rotation has no fixed points, so the quotient will have genus 2, and therefore this description is at this point less suitable for constructing these surfaces. Below are two examples of this case.
The first indicates a singly periodic Scherk surface with 8 ends as a limit, the second a simple pattern of noded planes, allowing Traizet regeneration. The general case allows for more complicated limits.
Resources
PoVRay sources
Minimal Surfaces 