The relativistically invariant coupling structures realized in a quark model with broken S${\mathrm{U}}_{6}$ \ifmmode\times\else\texttimes\fi{} ${\mathrm{O}}_{3}$ group symmetry are applied here to the calculation of the rare radiative decays of tensor mesons. In particular, we examine the decay modes $(f,{A}_{2})\ensuremath{\rightarrow}\ensuremath{\gamma}\ensuremath{\gamma},\ensuremath{\rho}\ensuremath{\gamma}, \mathrm{and} \ensuremath{\omega}\ensuremath{\gamma}$, based on the additional requirement of vector-meson dominance of the electromagnetic interactions of hadrons. Calculated partial decay rates are compared with previous estimates.