In single-component theories of dark matter, the 2 → 2 amplitudes for dark-matter production, annihilation, and scattering can be related to each other through various crossing symmetries. These crossing relations lie at the heart of the celebrated complementarity which underpins different existing dark-matter search techniques and strategies. In multi-component theories of dark matter, by contrast, there can be many different dark-matter components with differing masses. This then opens up a new, "diagonal" direction for dark-matter complementarity: the possibility of darkmatter decay from heavier to lighter dark-matter components. In this work, we discuss how this new direction may be correlated with the others, and demonstrate that the enhanced complementarity which emerges can be an important ingredient in probing and constraining the parameter spaces of such models.Introduction.-In recent years, many search techniques have been exploited in the hunt for dark matter [1]. These include possible dark-matter production at colliders; direct detection of cosmological dark matter through its scattering off ordinary matter at underground experiments; and indirect detection of dark matter through observation of the remnants of the annihilation of cosmological dark matter into ordinary matter at terrestrial or satellite-based experiments. At first glance, these different techniques may seem to rely on three independent properties of dark matter, namely its amplitudes for production, scattering, and annihilation. However, these three amplitudes are often related to each other through various crossing symmetries. As a result, the different corresponding search techniques are actually correlated with each other through their dependence on a single underlying interaction which couples dark matter to ordinary matter, and the results achieved through any one of these search techniques will have immediate implications for the others as well as for this underlying interaction. This is the origin of the celebrated complementarity which connects the different existing dark-matter search techniques (for a review, see Ref.[2]).