The general results of Ciabattoni & Ramanayake about display calculi are applied to the framework of atomic logics. We obtain an instantiation of their “I2 acyclic” formulas in this framework. Then, we show that our versions of I2 acyclic formulas are inductive formulas in the sense of Goranko & Vakarelov, again adapted to this framework. This allows us to provide a characterization of displayable calculi extending basic atomic logics in terms of these formulas: a calculus is properly displayable iff it is sound and complete w.r.t. a class of frames defined by I2 acyclic formulas. As corollary of the admissibility of the cut rule in display calculi, we obtain sufficient conditions for obtaining conservative extensions of logics. Then, we give examples of correspondences between structural inference rules and first-order frame properties and we detail some of the computations that enable to transform one to the other. Finally, we apply our characterization theorem to propositional logic, FDE, the family of relevance logics, intuitionistic and intermediate logics and obtain novel calculi and conservativity results for these logics. In particular, we show how classical and standard inference rules reappear as simplifications of more general and abstract rules that stem from the display calculi of our atomic logics.