There exists a growing class of dinuclear complexes with bridging radical-anion ligands that is of interest both for bioinorganic and for supermolecular chemistry. Their bonding situation as well as chemical and spectroscopic properties are not described adequately by standard models such as the ligand-field theory. For rational design of complexes with desired properties, it is thus necessary to understand better the interrelations between electronic structure, spin density, and electron paramagnetic resonance (EPR) parameters in dinuclear systems with redox-active bridging ligands and to evaluate the performance of density functional methods in their description. As particularly suitable, experimentally well-characterized representatives, a series of dinuclear copper(l) complexes with azo or tetrazine bridge ligands have been studied here by different density functional methods. To reproduce the available experimental metal hyperfine couplings, the inclusion of spin-orbit effects into the calculations is necessary. An unusual direction of the dependence of computed hyperfine couplings on an exact-exchange admixture into the exchange-correlation functional may be understood from a McConnell-type spin polarization of the a-framework of the bridge. Ligand nitrogen hyperfine couplings are also compared with experiment where available. Electronic g-tensors are reproduced well by the calculations and have been analyzed in detail in terms of atomic spin-orbit contributions and electronic excitations.