Computational text-level discourse analysis mostly happens within Rhetorical Structure Theory (RST), whose structures have classically been presented as constituency trees, and relies on data from the RST Discourse Treebank (RST-DT); as a result, the RST discourse parsing community has largely borrowed from the syntactic constituency parsing community. The standard evaluation procedure for RST discourse parsers is thus a simplified variant of PARSEVAL, and most RST discourse parsers use techniques that originated in syntactic constituency parsing. In this article, we isolate a number of conceptual and computational problems with the constituency hypothesis. We then examine the consequences, for the implementation and evaluation of RST discourse parsers, of adopting a dependency perspective on RST structures, a view advocated so far only by a few approaches to discourse parsing. While doing that, we show the importance of the notion of headedness of RST structures. We analyze RST discourse parsing as dependency parsing by adapting to RST a recent proposal in syntactic parsing that relies on head-ordered dependency trees, a representation isomorphic to headed constituency trees. We show how to convert the original trees from the RST corpus, RST-DT, and their binarized versions used by all existing RST parsers to head-ordered dependency trees. We also propose a way to convert existing simple dependency parser output to constituent trees. This allows us to evaluate and to compare approaches from both constituent-based and dependency-based perspectives in a unified framework, using constituency and dependency metrics. We thus propose an evaluation framework to compare extant approaches easily and uniformly, something the RST parsing community has lacked up to now. We can also compare parsers’ predictions to each other across frameworks. This allows us to characterize families of parsing strategies across the different frameworks, in particular with respect to the notion of headedness. Our experiments provide evidence for the conceptual similarities between dependency parsers and shift-reduce constituency parsers, and confirm that dependency parsing constitutes a viable approach to RST discourse parsing.