In most multicomponent dark-matter scenarios, two classes of processes generically contribute to event rates at experiments capable of probing the nature of the dark sector. The first class consists of "diagonal" processes involving only a single species of dark-matter particle-processes analogous to those which arise in single-component dark-matter scenarios. By contrast, the second class consists of "off-diagonal" processes involving dark-matter particles of different species. Such processes include inelastic scattering at direct-detection experiments, asymmetric production at colliders, dark-matter co-annihilation, and certain kinds of dark-matter decay. In typical multicomponent scenarios, the contributions from diagonal processes dominate over those from off-diagonal processes. Unfortunately, this tends to mask those features which are most sensitive to the multicomponent nature of the dark sector. In this paper, by contrast, we point out that there exist natural, multicomponent dark-sector scenarios in which the off-diagonal contributions actually dominate over the diagonal. This then gives rise to a new, enhanced picture of dark-matter complementarity. In this paper, we introduce a scenario in which this situation arises and examine the enhanced picture of dark-matter complementarity which emerges.