AbstractWe review Lie polynomials as a mathematical framework that underpins the structure of the socalled double copy relationship between gauge and gravity theories (and a network of other theories besides). We explain how Lie polynomials naturally arise in the geometry and cohomology of $$\mathcal {M}_{0,n}$$
M
0
,
n
, the moduli space of n points on the Riemann sphere up to Mobiüs transformation. We introduce a twistorial correspondence between the cotangent bundle $$T^*_D\mathcal {M}_{0,n}$$
T
D
∗
M
0
,
n
, the bundle of forms with logarithmic singularities on the divisor D as the twistor space, and $$\mathcal {K}_n$$
K
n
the space of momentum invariants of n massless particles subject to momentum conservation as the analogue of space–time. This gives a natural framework for Cachazo He and Yuan (CHY) and ambitwistorstring formulae for scattering amplitudes of gauge and gravity theories as being the corresponding Penrose transform. In particular, we show that it gives a natural correspondence between CHY halfintegrands and scattering forms, certain $$n3$$
n

3
forms on $$\mathcal {K}_n$$
K
n
, introduced by ArkaniHamed, Bai, He and Yan (ABHY). We also give a generalization and more invariant description of the associahedral $$n3$$
n

3
planes in $$\mathcal {K}_n$$
K
n
introduced by ABHY.