We investigate the classification of self-dual nonlinear electrodynamic (NED) theories based on their analyticity properties, which are directly linked to invariance under a discrete φ-parity transformation. This classification is expressed through the structure of the irrelevant \( T\overline{T} \) -like deformations that generate the theories from a Maxwell seed. Using both closed-form and perturbative methods within the Courant-Hilbert (CH) and Russo-Townsend auxiliary field formalisms, we demonstrate a precise correspondence: φ-parity-invariant, analytic theories are generated by irrelevant deformations built from integer powers of the energy-momentum tensor scalars, \( {\mathcal{O}}_{\lambda } \) ~ Σ Cm(TμνT μν)1−m(TμμTνν)m. Conversely, φ-parity-violating, non-analytic theories require deformations involving both integer and half-integer powers, \( {\mathcal{O}}_{\lambda } \) ~ Σ Cm(TμνT μν)1−m/2(TμμTνν)m/2. We prove this result in generality via a perturbative CH framework, showing that φ-parity invariance imposes specific constraints on the expansion coefficients of the CH function ℓ(τ) which, in turn, force all half-integer powers in the deformation to vanish. The classification is explicitly verified for known closed-form theories: the analytic generalized Born-Infeld model and the non-analytic examples of the q = 3/4-deformed and “no τ-maximum” theories. Furthermore, we show how the φ-parity transformation is consistently generalized in the presence of a marginal root- \( T\overline{T} \) coupling γ, and we derive the corresponding marginal and irrelevant flow equations for the studied theories.
