  We investigate configuration-space integrals over punctured Riemann spheres
from the viewpoint of the motivic Galois coaction and double-copy structures
generalizing the Kawai-Lewellen-Tye relations in string theory. For this
purpose, explicit bases of twisted cycles and cocycles are worked out whose
orthonormality simplifies the coaction. We present methods to efficiently
perform and organize the expansions of configuration-space integrals in the
inverse string tension $\alpha'$ or the dimensional-regularization parameter
$\epsilon$. Generating-function techniques open up a new perspective on the
coaction of multiple polylogarithms in any number of variables and analytic
continuations in the unintegrated punctures. We present a compact recursion for
a generalized KLT kernel and discuss its origin from intersection numbers of
Stasheff polytopes and its implications for correlation functions of
two-dimensional conformal field theories. We find a non-trivial example of
correlation functions in $(\mathfrak{p},2)$ minimal models, which can be
normalized to become uniformly transcendental in the $\mathfrak{p} \to \infty$
limit.
