I will discuss a construction associating a diagram to an irregular meromorphic connection on the Riemann sphere, motivated by the perspective of a classification of moduli spaces of meromorphic connections, i.e. wild character varieties. An essential feature of the construction is that the same diagram can be read in different ways, as coming from meromorphic connections with different irregular types, leading to isomorphisms between the corresponding moduli spaces. There are also Weyl group actions on the diagrams which match with the Okamoto symmetries of Painlevé equations. After reviewing this story for the already known cases, I will present results of ongoing work on how to extend part of this to the more general case, to connections having several irregular singularities.