Metasurfaces enable powerful control of electromagnetic waves using subwavelength planar structures, but their deeply subwavelength periodicity typically suppresses propagating diffraction orders, which limits the number of available scattering channels. Diffraction gratings and metagratings overcome this limitation by supporting multiple propagating diffraction orders, thus providing additional degrees of freedom for controlling wave propagation. However, when several diffraction channels are present, it becomes nontrivial to predict how spatial symmetries combined with reciprocity affect the overall scattering response. For this purpose we develop a formalism to determine the scattering symmetries of diffraction gratings supporting multiple diffraction orders. The approach is based on constructing a global scattering matrix that connects all incident and scattered diffraction channels and on introducing matrix representations of spatial symmetry operations acting on the field amplitudes. From these representations we derive an invariance condition that directly constrains the subscattering matrices associated with each pair of diffraction orders. This provides a rigorous approach for computing the grating scattering coefficients imposed by symmetry and reciprocity. We illustrate the application of this approach via several examples and show how metagratings may be used to achieve, for instance, angle-asymmetric transmission and extrinsic chiral effects.