Classical cumulants linearise properties of random variables normally encoded by moments. They are recursively defined through their generating function, as the logarithm of the corresponding characteristic function. For càdlàg semimartingales, the Marcus signature lift plays a similar role. Signature cumulants, defined as logarithm of expected signatures, are seen to satisfy a fundamental functional relation. This equation, in a deterministic setting, contains Hausdorff's differential equation, which itself underlies Magnus' expansion. The (commutative) case of multivariate cumulants arises as another special case and yields a new Riccati-type relation valid for general semimartingales. Here, the accompanying expansion provide a new view on recent "diamond" and "martingale cumulants" (Alos et al '17, Lacoin et al '19., Friz et al. '20) expansions. Some concrete examples are given.