We extend the potential-based formulation of self-dual and duality-symmetric gauge theories developed in to curved spacetime. The gravitational coupling is encoded by the same metric-dependent linear map of Sen’s formalism. The parent new action retains the higher-form $h$-gauge symmetry, and different gauge choices reproduce either Sen’s flux-based formulation or a potential-based description in which the shadow sector decouples directly at the level of the action. We also derive explicit form of the gravitational map for the toroidally reduced theory in $D=4n$ and for a polyform formulation in arbitrary dimension, both of which apply equally to ours as well as Sen’s formalism. As non-trivial checks, we reproduce the 'Alvarez-Gaum'e-Witten gravitational anomalies of the two-dimensional chiral boson without fermionisation, and of the ten-dimensional self-dual five-form. Finally, on ${AdS}_5\times S^5$ the physical on-shell action yields the non-vanishing boundary contribution required by holography without adding a boundary term by hand. We also establish a local on-shell identification with the textbook supergravity RR four-form potential.