We claim the answer is yes. Take the $4$ tangency points given by one of these tetrahedrons and its insphere, and call them $T_1, T_2, T_3, T_4$. Rotate these $4$ points around the bisector of $T_1 T_2$ and $T_3 T_4$ by $120$ and $240$ degrees. Our resulting $12$ points form a regular hexagon, and there exists isometries that map one of these polygons onto another. $\quad \blacksquare$