The Jorge-Meeks 4-Noid and the Tetrahedroid both belong to a 2-parameter family of 4-Noids that have two orthogonal symmetry planes. The parameters control the position of the ends and their relative growth rates.
These examples were described in Hermann Karcher’s Tokyo notes on pages 30ff.
Resources
Mathematica Notebook
PoVRay Sources
Minimal Surfaces 