The last decade has seen the development of path homology and magnitude homology -- two homology theories of directed graphs, each satisfying classic properties such as Kunneth and Mayer-Vietoris theorems. Recent work of Asao has shown that magnitude homology and path homology are related, appearing in different pages of a certain spectral sequence. Here we study the target of that spectral sequence, which we call reachability homology. We prove that it satisfies appropriate homotopy invariance, Kunneth, excision, and Mayer-Vietoris theorems, these all being stronger than the corresponding properties for either magnitude or path homology.