We prove that a 4d theory of non-linear electrodynamics has equations of motion which are equivalent to those of the Maxwell theory in curved spacetime, but with the usual metric gμν replaced by a unit-determinant metric hμν (F) which is a function of the field strength Fμν, if and only if the theory enjoys electric-magnetic duality invariance. Among duality-invariant models, the Modified Maxwell (Mod-Max) theory is special because the associated metric hμν (F) produces identical equations of motion when it is coupled to the Maxwell theory via two different prescriptions which we describe. We use the field-dependent metric perspective to analyze the electric and magnetic 1-form global symmetries in models of self-dual electrodynamics. This analysis suggests that any duality-invariant theory possesses a set of conserved currents jμ which are in one-to-one correspondence with 2-forms that are harmonic with respect to the field-dependent metric hμν (F).