Agol introduced veering triangulations of mapping tori, whose combinatorics are canonically associated to the pseudo-Anosov monodromy. In unpublished work, Guéritaud and Agol generalise an alternative construction to any closed manifold equipped with a pseudo-Anosov flow without perfect fits. Schleimer and I build the reverse map: from a transverse veering triangulation we canonically construct a dynamic pair in the sense of Mosher, which implies a combinatorial version of a pseudo-Anosov flow without perfect fits. I will also talk about work with Giannopolous and Schleimer building a census of transverse veering triangulations. The current census lists all transverse veering triangulations with up to 16 tetrahedra, of which there are 87,047. The number approximately doubles with each added tetrahedron.