<p>Gentzen-style sequent calculi are introduced for logics that integrate Abelian and connexive logics. These logics are referred to as Abelian connexive logics (ACLs). The proposed sequent calculi are obtained from Gentzen-style sequent calculi for Abelian group logic (AGL) by adding logical inference rules for connexive negations. Cut-elimination theorems are proved for the proposed calculi, along with embedding theorems into a Gentzen-style sequent calculus for AGL. Additionally, completeness theorems with respect to Abelian group semantics are proved for ACLs. Furthermore, it is demonstrated that ACLs are decidable using these calculi. Various negation-related properties of ACLs, such as contra-classicality, non-triviality, connexivity, negation-inconsistency, and irrelevancy, are investigated based on the calculi. Additionally, ACLs are identified as proper extensions of a fragment of extended classical linear logic with the structural rule of mix.</p>