Designing sparse directed spanners, which are subgraphs that approximately maintain distance constraints, has attracted sustained interest in TCS, especially due to their wide applicability, as well as the difficulty to obtain tight results. However, a significant drawback of the notion of spanners is that it does not capture a natural setting where demand pairs are subject to restrictions beyond a distance constraint. In this paper we initiate the study of **routing-controlled spanners**, where in addition to distance constraints, demand pairs are also subjected to routing constraints, which may require or forbid visiting specific vertices on feasible paths. The goal is to find a minimum-cost routing solution that satisfies the multiple constraints. Moreover, we introduce an even more general notion, which we call **packing-covering spanner**, where each demand pair is subjected to a number of packing and covering constraints, in addition to a distance constraint. Packing-covering spanners capture other natural network connectivity problems such as optimal hopsets and graph spanners for group Steiner distances. To the best of our knowledge, we obtain the first approximation algorithms for the packing-covering spanner problem, and thus for the routing-controlled spanners, under natural assumptions. Our results match the state-of-the-art approximation ratios in special cases of ours, such as Steiner Forests and Directed Spanners. Our results also imply approximation algorithms for optimal hopsets and graph spanners for group Steiner distances in the directed setting, and position packing-covering spanners as a natural abstraction unifying several well-studied problems in directed network design.