We take a fresh look at electromagnetic constitutive relations, questioning the standard approach that relates the fundamental electromagnetic fields E and B to the excitation fields D and H. Instead, in recognition of the fact that the excitation fields are not directly measurable, we set up a framework that relates the medium response directly to the fundamental fields. We do this in the simplest way, basing the relationship on first-order operators acting on E and B. This relationship, together with Faraday’s law, the absence of magnetic monopoles, and local charge conservation, forms the basis of our approach. The laws of Ampère and Gauss, which in the conventional macroscopic formulation of Maxwell’s equations contain D and H, are dispensed with. As well as reproducing conventional macroscopic electromagnetic theory in the appropriate limit, we show that this approach admits a richer gamut of possibilities of meta-media than does the conventional approach. In particular, we identify novel axion-like terms and discuss how these might be experimentally realized. Tracing our approach back to vacuum, we admit the possibility of an axion-like term augmenting the conventional vacuum constitutive relations. We show that it has a topological character, one that can be exploited by manipulation of the background spacetime manifold. In a simple model setting, we show that a careful examination of the mathematical basis of Stokes’ theorem leads to a severance of the link between local and global charge conservation.