We report calculations of the noncollinear magnetic structures of ${\mathrm{YFe}}_{12\ensuremath{-}x}{\mathrm{Mo}}_{x}$ permanent-magnet materials, using a novel variant of a spin-polarized tight-binding--linear-muffin-tin-orbital technique allowing for local spin-quantization axes on each site and considering spin-orbit coupling. The ternary ${\mathrm{YFe}}_{12\ensuremath{-}x}{\mathrm{Mo}}_{x}$ compounds crystallize in the tetragonal ${\mathrm{ThMn}}_{12}$ structure which can be stabilized only by the partial substitution of Fe by an early transition metal like Mo. We show that the substitutional disorder leads to canted spin structures at low Mo content ( $x\ensuremath{\sim}1$) and to spin-glass-like behavior at higher Mo content ( $x\ensuremath{\sim}3$).