This is the rhombic enneacontahedron, one of the few well-known zonohedra. Its ninety faces have two types: sixty wide rhombi, and thirty narrow rhombi.
In the image above, the thirty narrow rhombi of the rhombic enneacontahedron have been augmented with prisms.
The next step in today’s polyhedral play was to create the convex hull of this augmented rhombic enneacontahedron. This produced the solid shown immediately above. To make the one shown below, I next used a function called “try to make faces regular.” The result is a symmetrohedron with 122 faces: 12 regular pentagons, 30 rhombi, 60 almost-square isosceles trapezoids, and thirty equilateral triangles.
Finally, I examined the dual of this symmetrohedron, which turned out to have 120 faces: two sets of sixty kites each.
The program I used to create these polyhedral images is called Stella 4d, and you can try it yourself (as a free trial download) at http://www.software3d.com/Stella.php.
This combination of thick and thin rhombs is making me think of Penrose’s P3 tiling: https://en.wikipedia.org/wiki/Penrose_tiling#Rhombus_tiling_(P3) Are the angles the same?
Not quite, no.