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High-harmonic generation (HHG) in solids has been touted as a way to probe ultrafast dynamics and crystal symmetries in condensed matter systems. Here, in a combined theoretical and experimental effort, we illustrate that the polarization properties of high harmonics in monolayer MoS₂ encode the dynamical symmetry properties of the crystal and laser, as well as material-specific properties such as the vectorial character of the transition dipole moments from different valence-conduction-band pairs. Our findings and methodology can readily be generalized to other condensed-matter systems, and relevant for the materials community where this new spectroscopy is expected to find increasingly applications to uncover material properties.