  This paper presents a systematic cataloging of the generators of celestial
symmetries on phase space. Starting from the celestial OPEs, we first show how
to extract a representation of the general-spin analog of the wedge subalgebra
of $w_{1+\infty}$ on the phase space of massless matter fields of arbitrary
helicity. These generators can be expressed as light-sheet operators that are
quadratic in the matter fields at future or past null infinity. We next show
how to extend these symmetries beyond the wedge. Doing so requires us to
augment the quadratic operators with: 1) linear terms corresponding to primary
descendants of the negative helicity gauge fields the matter modes couple to,
and 2) a tower of higher-particle composite operator contributions. These modes
can be realized as light-ray operators supported on generators of null
infinity, but local on the celestial sphere. Finally, we construct a
representation of the celestial symmetries that captures how the positive
helicity gauge fields transform. We close by discussing how these celestial
symmetries inform our choice of detector operators.
