We report on acoustically driven spin resonances in atomic-scale centers in silicon carbide at room temperature. Specifically, we use a surface acoustic wave cavity to selectively address spin transitions with magnetic quantum number differences of $pm$1 and $pm$2 in the absence of external microwave electromagnetic fields. These spin-acoustic resonances reveal a non-trivial dependence on the static magnetic field orientation, which is attributed to the intrinsic symmetry of the acoustic fields combined with the peculiar properties of a half-integer spin system. We develop a microscopic model of the spin-acoustic interaction, which describes our experimental data without fitting parameters. Furthermore, we predict that traveling surface waves lead to a chiral spin-acoustic resonance, which changes upon magnetic field inversion. These results establish silicon carbide as a highly-promising hybrid platform for on-chip spin-optomechanical quantum control enabling engineered interactions at room temperature.