What is the name of the polyhedral shape of the Humanity Star?

Here I've put red dots on corners where six triangles meet and green dots on corners where five triangles meet:

This has too many red dots together to be the pentakis icosidodecahedron suggested by some commenters -- in particular, the two red-red-red triangles sharing an edge slightly below the middle of the image are not to be found in that polyhedron.

What it does look like to me is a pentakis icosidodecahedron that one has cut up along the one of the "equators" and then twisted one half by $36^\circ$ before gluing them together again.

This figure still has 80 sides rather than the claimed 65, though. But on the other hand, I can count 35 sides visible in this picture, and so many triangles should not be visible from one point in a somewhat regularly constructed 65-sided polyhedron anyway.

A systematic name for it would be a pentakis pentagonal orthobirotunda.