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Specific features of orbital and spin structure of transition metal compoundsin the case of the face-sharing MO$_6$ octahedra are analyzed. In thisgeometry, we consider the form of the spin--orbital Hamiltonian for transitionmetal ions with double ($e_g^{\sigma}$) or triple ($t_{2g}$) orbitaldegeneracy. Trigonal distortions typical of the structures with face-sharingoctahedra lead to splitting of $t_{2g}$ orbitals into an $a_{1g}$ singlet and$e_g^{\pi}$ doublet. For both doublets ($e_g^{\sigma}$ and $e_g^{\pi}$), in thecase of one electron or hole per site, we arrive at a symmetric model with theorbital and spin interaction of the Heisenberg type and the Hamiltonian ofunexpectedly high symmetry: SU(4). Thus, many real materials with this geometrycan serve as a testing ground for checking the prediction of this interestingtheoretical model. We also compare general trends in spin--orbital("Kugel--Khomskii") exchange interaction for three typical situations: those ofMO$_6$ octahedra with common corner, common edge, and the present case ofcommon face, which has not been considered yet.

Spin--orbital interaction for face-sharing octahedra: Realization of a highly symmetric SU(4) model