  We prove that every unitary invertible quantum field theory satisfies a
generalization of the famous spin statistics theorem. To formulate this
extension, we define a `higher spin' action of the stable orthogonal group $O$
on appropriate spacetime manifolds, which extends both the reflection
involution and spin flip. On the algebraic side, we define a `higher
statistics' action of $O$ on the universal target for invertible field
theories, $I\mathbb{Z}$, which extends both complex conjugation and fermion
parity $(-1)^F$. We prove that every unitary invertible quantum field theory
intertwines these actions.
