  We show that the form of the recently proposed subleading soft graviton and
gluon theorems in any dimension are severely constrained by elementary
arguments based on Poincar\'e and gauge invariance as well as a
self-consistency condition arising from the distributional nature of scattering
amplitudes. Combined with the assumption of a local form as it would arise from
a Ward identity the orbital part of the subleading operators is completely
fixed by the leading universal Weinberg soft pole behavior. The polarization
part of the differential subleading soft operators in turn is determined up to
a single numerical factor for each hard leg at every order in the soft momentum
expansion. In four dimensions, factorization of the Lorentz group allows to fix
the subleading operators completely.
