Classical Core Logic (CCL) is a system put forward by Tennant (2017) for relevant deductive reasoning. Like many relevant logics, CCL achieves relevance by restricting the structural rule of Weakening. Unlike relevant logics in the R family, it blocks the Lewis argument for explosion not by rejecting Disjunctive Syllogism, but by rejecting unrestricted Cut. Although CCL deserves the title of substructural logic, if only because it restricts Weakening and Cut, this remains a rather superficial characterization. It is often pointed out that interesting substructural logics need to preserve the operational rules of the logic the structural rules of which are being restricted. On this count, CCL behaves in a rather unusual way, by requiring unassorted pairs of connectives (i.e., multiplicative on one side, additive on the other). The goal of this paper is to bring out the unusual features of Classical Core Logic qua substructural logic, both from a proof-theoretical and a semantical point of view, and provide a principled explanation for them, based on the specific understanding of the notion of relevance that CCL explicates. The general picture that emerges from all this is a commitment to a specific form of premise/conclusion relativism, i.e., the view according to which connectives may receive different interpretations depending on whether they occur as premises or conclusions in a sequent. This is joint work with Peter Verdée from the Catholic University of Louvain-la-Neuve.