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.