Some years ago Walter Carnielli and his colleagues developed a novel way of approaching paraconsistency. They introduced primitive operators to handle the notions of consistency and inconsistency, and so they created a whole new family of logical systems: the logics of formal inconsistency (LFI). In this paper we present a conceptual characterization of two of the LFI’s most important operators, the so-called consistency and inconsistency operators. Based on the relation between the meaning of logical constants and the structural properties of inferences, we show that these connectives may be understood as devices that recover the ‘hidden symmetry of Logic’ expressed by structural negation. To analyze how structural and operational rules interact, we designed a sequent calculus for LFI1 and LFI2, based on a method also introduced by Carnielli.