Could graviton exist in a single polarisation?

Penrose’s twistor theory can only construct a left-handed graviton. This is seen as a problem.

But… is there anything wrong with gravitons existing in only one polarisation?

How would gravity differ if this were the case.

For example, would it affect photons of different polarisations differently as they are bent by a star?

Would it effect the angular momentum of black holes?

Since we have never seen a graviton, we have no idea if they appear in more than one polarisation like the photon.

Presumably we would not be able to construct GR with only one polarisation of gravitons. But perhaps, if we abandon the right-handed part of GR and keep only the left-handed part, this might still predict all the gravitational effects we know yet may solve some other problems that Twistor theory sets out to solve.

So, is left-handed gravitation consistent with observation?