The skeletal graphs of Alan Schoen’s T’-R’ surface have horizontal layers based on triangles and rhombi. The translational fundamental piece below has reflectional symmetry edges that lie on the faces of a prism over a regular hexagon. This surface has genus 6.
The surface comes in a 1-parameter family. In both limits the height of the prism shrinks to 0. In one case, one obtains a nodal limit with horizontal planes joined by catenoidal necks:
The other limit features 12-ended Scherk surfaces.
Resources
Mathematica Notebook
PoVRay Sources
Minimal Surfaces 