  We present a new formula for the coaction of a large class of integrals. When
applied to one-loop (cut) Feynman integrals, it can be given a diagrammatic
representation purely in terms of pinches and cuts of the edges of the graph.
The coaction encodes the algebraic structure of these integrals, and offers
ways to extract important properties of complicated integrals from simpler
functions. In particular, it gives direct access to discontinuities of Feynman
integrals and facilitates a straightforward derivation of the differential
equations they satisfy, which we illustrate in the case of the pentagon.
