We provide a new characterization of the NP-hard arc routing problem Rural Postman in terms of a constrained variant of minimum-weight perfect matching on bipartite graphs. To this end, we employ a parameterized equivalence between Rural Postman and Eulerian Extension, a natural arc addition problem in directed multigraphs. We indicate the NP-hardness of the introduced matching problem. In particular, we use the matching problem to make partial progress towards answering the open question about the parameterized complexity of Rural Postman with respect to the parameter "number of weakly connected components in the graph induced by the required arcs". This is a more than thirty years open and long-neglected question with significant practical relevance.