We argue in this article that many common adverbial phrases generally taken to signal a discourse relation between syntactically connected units within discourse structure instead work anaphorically to contribute relational meaning, with only indirect dependence on discourse structure. This allows a simpler discourse structure to provide scaffolding for compositional semantics and reveals multiple ways in which the relational meaning conveyed by adverbial connectives can interact with that associated with discourse structure. We conclude by sketching out a lexicalized grammar for discourse that facilitates discourse interpretation as a product of compositional rules, anaphor resolution, and inference.