Cells are identical trigonal trapezohedra or rhombohedra.
Conway, Burgiel, and Goodman-Strauss call it an oblate cubille.
It is analogous to the regular hexagonal being dissectable into 3 rhombi and tiling the plane as a rhombille.
The rhombille tiling is actually an orthogonal projection of the trigonal trapezohedral honeycomb.
A different orthogonal projection produces the quadrille where the rhombi are distorted into squares.