 Solution

The $z$ axis is a $C_2$ axis through the centers of opposite cube faces: a half turn permutes atoms within each species. A body diagonal through sites 1 and 8 is a $C_3$ axis: a third turn fixes those two opposite vertices and cycles the other three vertices of each tetrahedral sublattice. These are symmetry operations of the tetrahedral symmetry group $T_d$; a quarter turn about $z$ would interchange the two chemical species and is not a symmetry of the specified cluster.