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States with core vacancies, which are commonly created by absorption of X-ray photons, can decay by a two-electron process in which one electron fills the core hole and the second one is ejected. These processes accompany many X-ray spectroscopies. Depending on the nature of the initial core-hole state and the decay valence-hole states, these processes are called Auger decay, intermolecular Coulomb decay, or electron-transfer-mediated decay. To connect many-body wavefunctions of the initial and final states with molecular orbital picture of the decay, we introduce a concept of natural Auger orbitals (NAOs). NAOs are obtained by two-step singular value decomposition of the two-body Dyson orbitals, reduced quantities that enter the expression of the decay rate in the Feshbach--Fano treatment. NAOs afford chemical insight and interpretation of the high-level ab intio calculations of Auger decay and related two-electron relaxation processes.