The windy postman problem is the NP-hard problem of finding the minimum cost of a tour traversing all edges of an undirected graph, where the cost of traversal of an edge depends on the direction. Given an undirected graph $G$, we consider the polyhedron $O(G)$ induced by the linear programming relaxation of a well-known integer programming formulation of the problem. We say that $G$ is windy postman perfect if $O(G)$ is integral. There exists a polynomial-time algorithm, based on the ellipsoid method, to solve the windy postman problem for the class of windy postman perfect graphs. Eulerian graphs and trees are windy postman perfect. By considering a family of polyhedra related to $O(G)$, we prove that series-parallel graphs are windy postman perfect, therefore solving a conjecture of [Win1987a].