  Gravitational interferometers and cosmological observations of the cosmic
microwave background offer us the prospect to probe the laws of gravity in the
primordial universe. To study and interpret these datasets we need to know the
possible graviton non-Gaussianities. To this end, we derive the most general
tree-level three-point functions (bispectra) for a massless graviton to all
orders in derivatives, assuming scale invariance. Instead of working with
explicit Lagrangians, we take a bootstrap approach and obtain our results using
the recently derived constraints from unitarity, locality and the choice of
vacuum. Since we make no assumptions about de Sitter boosts, our results
capture the phenomenology of large classes of models such as the effective
field theory of inflation and solid inflation. We present formulae for the
infinite number of parity-even bispectra. Remarkably, for parity-odd bispectra,
we show that unitarity allows for only a handful of possible shapes: three for
graviton-graviton-graviton, three for scalar-graviton-graviton and one for
scalar-scalar-graviton, which we bootstrap explicitly. These parity-odd
non-Gaussianities can be large, for example in solid inflation, and therefore
constitute a concrete and well-motivated target for future observations.
