<p>We consider properties of non-linear theories of a chiral 4-form gauge field <i>A</i>₄ in ten space-time dimensions with an emphasis on a subclass of these theories which are invariant under the <i>D</i> = 10 conformal symmetry. We show that general results regarding a peculiar structure of duality-invariant abelian gauge theories in four and six space-time dimensions do not extend to non-linear chiral 4-form theories in ten dimensions. This discrepancy arises primarily from the large number 81 of independent invariants constructible from the self-dual part of the five-form field strength <i>F</i>₅ = <i>dA</i>₄ in <i>D</i> = 10, in stark contrast to the single independent (fourth-order) invariants which are building blocks of the actions in the lower dimensional cases. In particular, unlike the <i>D</i> = 4 and <i>D</i> = 6 cases, where non-linear duality-invariant theories can be viewed as stress-tensor (<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 of seed theories, the flow equations in <i>D</i> = 10 generally involve both stress-tensor invariants and additional higher-order structures constructed from <i>F</i>₅. In passing, we prove the equivalence of three Lagrangian formulations of non-linear duality-invariant <i>p</i>-form theories: the PST, the Ivanov-Nurmagambetov-Zupnik and the “clone” one.</p>