What is two tetrahedrons stuck together called?

Two regular tetrahedrons connected face to face make a "regular triangular dipyramid."

That is one of the 92 "Johnson solids." Those are the convex polyhedrons whose faces are regular polygons but do not belong to either of the two sets of highly symmetric polyhedrons (the Platonic and the Archimedean) or to the perhaps less interesting two infinite families of prisms and antiprisms.

If the two tetrahedrons overlap, both centers at the same place but with the tetrahedrons facing in opposite directions, it makes a "stellated octahedron."