We use symmetry analysis and density functional theory to show that changes in magnetic order at a surface with respect to magnetic order in the bulk can be generically determined by considering local magnetoelectric responses of the crystal. Specifically, analysis of the atomic-site magnetoelectric responses, or equivalently the corresponding local magnetic multipoles, can be used to predict all surface magnetic modifications arising purely from symmetry lowering via termination of the bulk magnetic order. This analysis applies even in materials with no bulk magnetoelectric response or surface magnetization. We then demonstrate our arguments for two example antiferromagnets, metallic CuMnAs and rock-salt NiO. We find that the (010) and (1-10) surfaces of CuMnAs and NiO respectively exhibit a series of antiferroically, as well as roughness-sensitive, ferroically ordered, modifications of the surface magnetic dipole moments, via canting or changes in sublattice magnitude, consistent with the bulk ordering of the magnetic multipoles. Our findings demonstrate a universal bulk-boundary correspondance allowing the general prediction of minimal possible surface and interface magnetic modifications, even in non-magnetoelectric materials. Furthermore, it paves the way for more accurate interpretations of a wide variety of surface-sensitive measurements.