Neumann's principle states if a crystal is invariant with respect to certain symmetry elements, any of its physical properties must also be invariant with respect to the same symmetry elements.
We typically use this principle to investigate the form of the tensor of physical properties. There are countless examples, for example conductivity tensor relating electromagnetic fields and current responses. However, all examples I have seen are relating macroscopic quantities.
Can we use the same principle to relate microscopic quantities? Examples would be spin-orbit or spin-spin correlations. I suppose we could use exactly the same argument. The problem would however be that the microscopic quantities are local in nature, they have are position dependent and somehow transforms under translations. In contrast currents and electromagnetic fields are not necessarily local, but may only have directions.