Celestial <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>L</mml:mi> <mml:msub> <mml:mi>w</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:msub> </mml:math> symmetries a

Pursuing our analysis of the companion Letter, [Celestial symmetries of black hole horizons, .], we study the gravitational solution space around a null hypersurface in the bulk of spacetime, such as a black hole or a cosmological horizon. We discuss the corresponding characteristic initial value problem both in the metric and Newman-Penrose formalisms and establish an explicit dictionary between the two. This allows us to identify Weyl-covariant structures in the solution space, including hierarchies of recursion relations encoding the flux-balance laws. We then establish a correspondence between the gravitational phase space at null infinity and the subleading phase space around the null hypersurface at finite distance. This connection is naturally formulated within the Newman-Penrose formalism by performing a partially off-shell conformal compactification and identifying the analog of the Ashtekar-Streubel symplectic structure in the radial expansion near the null hypersurface. Using this framework, we identify the celestial <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>L</a:mi> <a:msub> <a:mi>w</a:mi> <a:mrow> <a:mn>1</a:mn> <a:mo>+</a:mo> <a:mi>∞</a:mi> </a:mrow> </a:msub> </a:math> symmetries in the subleading phase space at finite distance by constructing their canonical generators and imposing self-duality conditions. This allows us to define a notion of covariant radiation, whose absence gives rise to an infinite tower of conserved charges, revealing physical quantities relevant to observers near black hole or cosmological horizons. As a concrete illustration, we consider the case of the self-dual Taub-Newman-Unti-Tamburino black hole.