A nice person called Payton Asch sent me an email with an observation about the Herschel enneahedron:
It looks like the underlying polytope for the enneahedron is a triangular bipyramid (two tetrahedra stacked on top of each other) or the dual polytope would be a triangular prism.
In the case of the triangular bipyramid you would truncate each of the vertices around the “equator” deep enough until the truncated areas meet at a vertex.