  All 4D gauge and gravitational theories in asymptotically flat spacetimes
contain an infinite number of non-trivial symmetries. They can be succinctly
characterized by generalized 2D currents acting on the celestial sphere. A
complete classification of these symmetries and their algebras is an open
problem. Here we construct two towers of such 2D currents from
positive-helicity photons, gluons, or gravitons with integer conformal weights
$k=1,0,-1,...$. These generate the symmetries associated to an infinite tower
of conformally soft theorems. The current algebra commutators are explicitly
derived from the poles in the OPE coefficients, and found to comprise a rich
closed subalgebra of the complete symmetry algebra.
