  We point out that tensor consistency relations-i.e. the behavior of
primordial correlation functions in the limit a tensor mode has a small
momentum-are more universal than scalar consistency relations. They hold in the
presence of multiple scalar fields and as long as anisotropies are diluted
exponentially fast. When de Sitter isometries are approximately respected
during inflation this is guaranteed by the Higuchi bound, which forbids the
existence of light particles with spin: De Sitter space can support scalar hair
but no curly hair. We discuss two indirect ways to look for the violation of
tensor con- sistency relations in observations, as a signature of models in
which inflation is not a strong isotropic attractor, such as solid inflation:
(a) Graviton exchange contribution to the scalar four-point function; (b)
Quadrupolar anisotropy of the scalar power spectrum due to super-horizon tensor
modes. This anisotropy has a well-defined statistics which can be distinguished
from cases in which the background has a privileged direction.
