It has the same vertices and edges as the regular octahedron. It also shares 4 of the 8 triangular faces of the octahedron, but has three additional square faces.
It is the only non-prismatic uniform polyhedron with an odd number of faces.
It is a non-orientable surface. It is unique as the only uniform polyhedron with an Euler characteristic of 1 and is hence a representation of the real projective plane very similar to the Roman surface.
The "hemi" part of the name means some of the faces form a group with half as many members as some regular polyhedron - here, three square faces form a group with half as many faces as the regular hexahedron, better known as the cube - hence -hemihexahedron. Hemi faces are also oriented in the same direction as the regular polyhedron's faces. The three square faces of the tetrahemihexahedron are, like the three facial orientations of the cube, mutually perpendicular.
The "half-as-many" characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. Visually, each square is divided into four right triangles, with two visible from each side.