The relation between two branches of solutions (radiative and subradiative) of wave equations on Minkowski spacetime is investigated, for any integer spin, in flat Bondi coordinates where remarkable simplifications occur and allow for exact boundary-to-bulk formulae. Each branch carries a unitary irreducible representation of the Poincare group, though an exotic one for the subradiative sector. These two branches of solutions are related by an inversion and, together, span a single representation of the conformal group. While radiative modes are realised in the familiar holographic way (either as boundary data at null infinity or as bulk fields with radiative asymptotic behavior), the whole tower of subradiative modes forms an indecomposable representation of the usual Poincare group, which can be encoded into a single boundary field living on an interior null cone. Lorentz transformations are realised in both cases as conformal transformations of the celestial sphere. The vector space of all subradiative modes carries a unitary representation of a group isomorphic to the Poincare group, where bulk conformal boosts play the role of bulk translations.