Alan Schoen’s Complementary H surfaces has the same pattern of straight lines and reflectional symmetries as the Schwarz H surface but adds a few handles, increasing the genus to 7. Below is a translational fundamental piece of a member of the 1-parameter family as shown in Schoen’s Nasa report.
The different assembly below shows the same surface and several replicates, the latter from the top, for esthetic appeal.
In one direction, the triangular holes widen and we obtain Callahan-Hoffman-Meeks surfaces with dihedral symmetry of order 3 as the limit.
In the other direction, we pull the triangles first apart,
until we reach a simpler nodal limit.
Little is known about such complementary configurations in general. Below is an animation of the 1-parameter deformation.