Here we have k-Noids with the symmetries of bipyramids over regular k-gons. This leaves a free parameter to adjust the relative growth rates of the ends with horizontal limiting normal versus the two ends with vertical normal.
For k=4, we have as a special case the octahedroid, which also is an antiprismatic 6-Noid.
There is little hope of classifying all k-Noids for large k, but these surfaces have a strong esthetic appeal.
Resources
PoVRay Sources
Minimal Surfaces 