<p>Auxiliary field techniques have recently gained interest in four-dimensional non-linear electrodynamics and two-dimensional integrable sigma models. In these settings, coupling a suitable “seed” theory to auxiliary fields provides a powerful mechanism to generate infinite families of models while preserving key dynamical properties, such as electromagnetic duality invariance in four dimensions and classical integrability in two dimensions. Deformations induced through auxiliary fields are closely related to <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math display="inline"> <mi>T</mi> <mover accent="true"> <mi>T</mi> <mo stretchy="true">¯</mo> </mover> </math></EquationSource> <EquationSource Format="TEX">\( T\overline{T} \)</EquationSource> </InlineEquation>-like deformations and, in two dimensions, also to their higher-spin generalisations. In this paper, we analyse and clarify the relations between different auxiliary field formulations in two and four dimensions, showing how they are governed by Legendre transformations of the interaction functions combined with appropriate field redefinitions. In four-dimensional electrodynamics, we establish a correspondence between the auxiliary field model of Russo and Townsend and the Ivanov-Zupnik formalism. In two dimensions, we develop the analogue of the Ivanov-Zupnik <i>μ</i>-frame to deform Principal Chiral, symmetric-space, non-Abelian T-dual, and (bi-)Yang-Baxter sigma models. We discuss how integrability is preserved and use properties of the <i>μ</i>-frame to further extend known families of integrable deformations.</p>