Bulletin of the Section of Logic. forthcomingThis article aims to study, proof-theoretically and semantically, Gentzen-style sequent calculi (including possibly Cut-free and Identity-free systems), containing combinations of canonical and cocanonical rules, i.e. Gentzen systems for sequents, with well-behaved forms of left and right introduction and elimination rules for logical expressions. Our main goals are to provide soundness and completeness results for the derivability relations of the