The magnetoreflectance of ${\mathrm{Hg}}_{1\ensuremath{-}x}{\mathrm{Cd}}_{x}\mathrm{Te}$ single crystals has been studied for alloy compositions $0.175\ensuremath{\le}x\ensuremath{\le}0.269$ in the small-gap semiconductor region, at sample temperatures of 24 and 91 K. The peaks in magnetoreflection spectra taken in the Faraday configuration are interpreted as transitions from Landau levels in the ${\ensuremath{\Gamma}}_{8}$ valence band to the ${\ensuremath{\Gamma}}_{6}$ conduction band. The peak positions were fit to transition energies calculated using the quasi-Ge model of Pidgeon and Brown. We find that only the energy gap ${E}_{g}$ is sensitive to alloy composition and temperature, the other parameters of this model showing no systematic variation within the experimental uncertainty. These parameters are ${E}_{p}=17.9\ifmmode\pm\else\textpm\fi{}0.2$ eV, ${\ensuremath{\gamma}}_{1}=2.5\ifmmode\pm\else\textpm\fi{}0.4$, ${\ensuremath{\gamma}}_{2}=\ensuremath{-}0.3\ifmmode\pm\else\textpm\fi{}0.2$, ${\ensuremath{\gamma}}_{3}=0.5\ifmmode\pm\else\textpm\fi{}0.1$, $\ensuremath{\kappa}=\ensuremath{-}1.2\ifmmode\pm\else\textpm\fi{}0.1$, and $F=\ensuremath{-}0.7\ifmmode\pm\else\textpm\fi{}0.3$. Using our parameters, we calculate the band-edge effective mass and $g$ factor in the conduction band as a function of alloy composition and temperature. The heavy-hole effective masses were determined to be 0.40, 0.45, and 0.53 for applied magnetic fields along the [001], [110], and [111] axes, respectively.