The double copy relationship between Yang-Mills theory and general relativity can be stated in terms of a field theory Kawai-Lewellen-Tye (KLT) momentum kernel, which maps two colour-ordered gluon amplitudes to a graviton amplitude at tree-level. These amplitudes can also be written in compact, helicity-graded representations on twistor space which include the famous Parke-Taylor and Hodges formulae in the maximal helicity violating sector. However, a double copy formulation of these helicity-graded formulae has proved elusive. In this paper, we use graph-theoretic methods to obtain an explicit double copy representation of the tree-level, helicity graded S-matrix of general relativity in terms of a KLT-like integral kernel in twistor space. This integral kernel glues together two colour-ordered integrands for tree-level gluon scattering on twistor space to produce tree-level graviton amplitudes, and admits a chiral splitting into positive and negative helicity degrees of freedom. Furthermore, the kernel can be inverted to obtain a new formula for the tree-level S-matrix of biadjoint scalar theory, which we verify using recursion relations. We also derive extensions of this integral kernel to graviton scattering in anti-de Sitter space and self-dual radiative spacetimes, commenting on their potential double copy interpretations.