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Abstract For any causal nonlinear electrodynamics theory that is “self-dual” (electromagnetic U(1)-duality invariant), the Legendre-dual pair of Lagrangian and Hamiltonian densities L H $$ \left\{\mathcal{L},\mathcal{H}\right\} $$ are constructed from functions ℓ h $$ \left\{\ell, \mathfrak{h}\right\} $$ on ℝ + related to a particle-mechanics Lagrangian and Hamiltonian. We show how a ‘duality’ relating ℓ to h $$ \mathfrak{h} $$ implies that L $$ \mathcal{L} $$ and H $$ \mathcal{H} $$ are related by a simple map between appropriate pairs of variables. We also discuss Born’s “Legendre self-duality” and implications of a new “Φ-parity” duality. Our results are illustrated with many examples.