Link to the least-lousy picture I could find of an example: http://www.steelpillow.com/polyhedra/icosa/tidystelfacet/sphere_cells.png
Only with each two parallel bands in the example replaced by single line.
It’s related to the icosahedron obviously and it’s symmetrical, but it’s not a semi-regular polyhedron since neither the triangles nor the hexagons are regular. I think it’s what you get when you take a truncated icosahedron (soccerball pattern) and then truncate it again. It may not be obvious from the picture, but it’s edges form ten great circles each dividing a sphere in half, and so it could be considered a family with the octahedron, cuboctahedron, and icosidodecahedron, which do the same thing with three, four, and six great circles. In fact they could be considered corresponding to the four Platonic solids that have three, four, six and ten opposing vertices and faces.
Anyway, as this long-winded explanation shows, it would really help if I could knew this beast’s name.