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 (
ρ
,
J
) that matter; and so we introduce appropriate linear operator equations that combine the Gauss and Maxwell-Ampère equations with the constitutive relations, eliminating (
D
,
H
). The result is that we can admit more types of electromagnetic media, notably, these relations can allow coupling in the bulk to a homogeneous axionic material; in contrast to standard electromagnetism where any homogeneous axionlike field is completely decoupled in the bulk, and only accessible at boundaries. We also consider a wider context, including the role of topology, extended nonaxionic constitutive parameters, and treatment of Ohmic currents. A range of examples including an axionic 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
).