Just as 3D cupolas can be thought of as the "expansion" of a polygon (start with the top face, expand its edges outward while moving downwards: it traces out the volume of the cupola), we can make 4D cupolas by starting with a cell and expanding it downwards, resulting in a polytope with the original polyhedron as the top cell, its cantellated counterpart as the bottom cell, and a bunch of prisms and other polyhedra in between.
For example, the tetrahedral cupola is formed by starting with the top tetrahedron, expanding it to a cuboctahedron (=runcinated tetrahedron), while filling in the gaps with triangular prisms and other tetrahedra. Specifically, the cupola consists of 5 tetrahedra, 10 triangular prisms, and a cuboctahedron.
Now a tetrahedral bicupola is formed by attaching two such cupola base-to-base. However, there are two ways of fitting the cupola together: either in such a way as to have 4 tetrahedra in one cupola share a face with 4 tetrahedra in the other cupola, which gives the orthobicupola (having bilateral symmetry), or rotated in complementary orientation so that no two tetrahedra share a face, which gives the gyrobicupola.
Now the latter, the tetrahedral gyrobicupola, has some rather interesting properties... which I'll refrain from stating for now, just to see if anyone else guesses it. :-) (Here's a hint: what's a triangular gyrobicupola?)