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?
 A: Recent observations at LIGO (GW170814 disfavour all non-standard assumptions for polarizations of gravitational wave, i.e. the observational data so far fits best to the standard concept of two ("plus" and "cross") possible polarizations.
A: Penrose's Non-linear graviton construction works for a complex Riemannian Manifold (dim - 4). Here, one deforms a twistor space (or its dual space) to produce anti self dual (resp. self dual) parts of Weyl curvature which satisfy source free graviton equation. However, we live in a real Lorentzian manifold so both left handed and right handed gravitons are complex conjugates of one another, one cannot exist without the other. To understand the individual impacts of a.s.d. and s.d. part of curvature, we would need to know how different spinor fields couples with these curvature components in a curved space-time, I guess spinor fields which are either left handed or right handed can be differentially influenced. So we would need chiral fields to understand chiral nature of gravity. Besides, polarization of gravitational waves and duality of complex weyl curvatures (as constructed using Twistors) are two different things.
In Palatial Twistor approach, one can construct a general source free Weyl curvature which can be restricted to a real Lorentzian manifold. The non-linear graviton construction forms the building block for palatial twistor approach.
