  We present an analytical model of orbital ordering in vanadium spinel MnV2O4.
The model is based on recent first-principles calculation indicating a strong
trigonal distortion at the vanadium sites of this compound [Phys. Rev. Lett.
102, 216405 (2009)]. At the single-ion level, the trigonal crystal field leaves
a doubly degenerate atomic ground state and breaks the approximate rotational
symmetry of t2g orbitals. We find that the effective interaction between the
low-energy doublets is described by a quantum antiferromagnetic 120-degree
model on the pyrochlore lattice. We obtain the classical ground state and show
its stability against quantum fluctuations. The corresponding orbital order
consisting of two inequivalent orbital chains is consistent with the
experimentally observed tetragonal symmetry. A periodic modulation of electron
density function along orbital chains is shown to arise from the staggering of
local trigonal axes. In the presence of orbital order, single-ion spin
anisotropy arising from relativistic spin-orbit interaction stabilizes the
experimentally observed orthogonal magnetic structure.
