  The diagrammatic coaction underpins the analytic structure of Feynman
integrals, their cuts and the differential equations they admit. The coaction
maps any diagram into a tensor product of its pinches and cuts. These
correspond respectively to differential forms defining master integrals, and
integration contours which place a subset of the propagators on shell. In a
canonical basis these forms and contours are dual to each other. In this talk I
review our present understanding of this algebraic structure and its
manifestation for dimensionally-regularized Feynman integrals that are
expandable to polylogarithms around integer dimensions. Using one- and two-loop
integral examples, I will explain the duality between forms and contours, and
the correspondence between the local coaction acting on the Laurent
coefficients in the dimensional regulator and the global coaction acting on
generalised hypergeometric functions.
