Determining a rotation of a tetrahedron from a projection

The following problem, arising from medical imaging, is addressed: Suppose that $T$ is a known tetrahedron in $\R^3$ with centroid at the origin. Also known is the orthogonal projection $U$ of the vertices of the image $\phi T$ of $T$ under an unknown rotation $\phi$ about the origin. Under what circumstances can $\phi$ be determined from $T$ and $U$?