  Motivated by recent theoretical and experimental controversy, we present a
theoretical study to clarify the orbital symmetry of the ground state of
vanadium spinel oxides AV$_2$O$_4$ (A=Zn, Mg, Cd). The study is based on an
effective Hamiltonian with spin-orbital superexchange interaction and a local
spin-orbit coupling term. We construct a classical phase-diagram and prove the
complex orbital nature of the ground state. Remarkably, with our new analysis
we predict correctly also the coherent tetragonal flattening of oxygen
octahedra. Finally, through analytical considerations as well as numerical
ab-initio simulations, we propose how to detect the predicted complex orbital
ordering through vanadium K edge resonant x-ray scattering.
