And, like these classical surfaces of Schwarz, it comes in a 1-parameter family that maintains its dihedral symmetries about a vertical axis, with a nodal limit and a Scherk limit.
The latter shows Alan Schoen’s reasoning for the name, because the two complementary regions have skeletal graphs based on a hexagon and a triangle, respectively. This also highlights a difference to P and H: The complementary regions are not congruent, as is the case for P, H, and in fact for any embedded genus 3 triply periodic minimal surface. H’-T has genus 4.
Two views of an intermediate member of this family are shown below.