  We revisit the standard construction of the celestial stress tensor as a
shadow of the subleading conformally soft graviton. In its original formulation
there is an obstruction to reproducing the expected TT OPE in the double soft
limit. We propose a modification to the definition which circumvents this
obstruction and then extend this change of basis beyond the conformally soft
and single helicity sectors. In the process we investigate how
(non)-commutativity of double soft limits is tied to the decoupling of primary
descendants, and how our choice of celestial basis determines which symmetries
are manifest at the level of the OPE beyond the MHV sector.
