I have two tetrahedra in $\mathbb R^3$, $T_1$ and $T_2$, and access to the coordinates of their vertices. $T_1$ and $T_2$ are tetrahedra in the sense that they each have four vertices, each vertex is connected to every other vertex, and no three edges lie along the same line. There are no further restrictions on their geometry.
Is there a simple method of determining whether the two tetrahedra are identical up to some rotation and translation operation? What if I wish to test whether $T_1$ and $T_2$ have some reflection (mirror) symmetry? One can, of course, exhaustively test for these things by attempting every possible mapping of $T_2$ onto $T_1$, which is my current strategy. However, I'm sure a better method must exist.