  We show how the standard constitutive assumptions for the macroscopic Maxwell
equations can be relaxed. This is done by arguing that the Maxwellian
excitation fields (D,H) should be dispensed with, on the grounds that they (a)
cannot be measured, and (b) act solely as gauge potentials for the charge and
current. In the resulting theory, it is only the links between the fields (E,B)
and the charge and current (\rho,J) that matter; and so we introduce
appropriate linear operator equations that combine the Gauss and Maxwell-Ampere
equations with the constitutive relations, eliminating (D,H). The result is
that we can admit more types of electromagnetic media -- notably, the new
relations can allow coupling in the bulk to a homogeneous axionic material; in
contrast to standard EM where any homogeneous axion-like field is completely
decoupled in the bulk, and only accessible at boundaries. We also consider a
wider context, including the role of topology, extended non-axionic
constitutive parameters, and treatment of Ohmic currents. A range of examples
including an axonic response material is presented, including static
electromagnetic scenarios, a possible metamaterial implementation, and how the
transformation optics paradigm would be modified. Notably, these examples
include one where topological considerations make it impossible to model using
(D,H).
