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We develop the theory of quantum (a.k.a. noncommutative) relations andquantum (a.k.a. noncommutative) graphs in the finite-dimensional covariantsetting, where all systems (finite-dimensional $C^*$-algebras) carry an actionof a compact quantum group $G$, and all channels (completely positive mapspreserving the canonical $G$-invariant state) are covariant with respect to the$G$-actions. We motivate our definitions by applications to zero-error quantumcommunication theory with a symmetry constraint. Some key results are thefollowing: 1) We give a necessary and sufficient condition for a covariantquantum relation to be the underlying relation of a covariant channel. 2) Weshow that every quantum confusability graph with a $G$-action (which we call aquantum $G$-graph) arises as the confusability graph of a covariant channel. 3)We show that a covariant channel is reversible precisely when its confusability$G$-graph is discrete. 4) When $G$ is quasitriangular (this includes allcompact groups), we show that covariant zero-error source-channel codingschemes are classified by covariant homomorphisms between confusability$G$-graphs.

Covariant quantum combinatorics with applications to zero-error communication