Based on ab initio pseudopotential calculations, we have examined the equilibrium atomic geometries and electronic structure of chemically abrupt epitaxial Al/GaAs and ${\mathrm{A}\mathrm{l}/\mathrm{G}\mathrm{a}}_{1\ensuremath{-}x}{\mathrm{Al}}_{x}\mathrm{As}(100)$ and (110) interfaces. In particular, we investigated the change in the corresponding Schottky barrier height for different interface atomic geometries and semiconductor alloy compositions. Our results indicate that different epitaxial geometries and orientations of the interface can change the absolute value of the Schottky barrier by as much as 0.4 eV. However, for a given equilibrium geometry of the interface, ${\mathrm{A}\mathrm{l}/\mathrm{G}\mathrm{a}}_{1\ensuremath{-}x}{\mathrm{Al}}_{x}\mathrm{As}(100)$ and (110) junctions exhibit compellingly similar barrier variations with alloy composition, which amount to the ${\mathrm{G}\mathrm{a}\mathrm{A}\mathrm{s}/\mathrm{G}\mathrm{a}}_{1\ensuremath{-}x}{\mathrm{Al}}_{x}\mathrm{As}$ band offset. The observed trend is explained on the atomic scale using a linear-response-theory approach.