Spin accommodation is demonstrated to play a determining role in the reactivity of silver cluster anions with oxygen. Odd-electron silver clusters are found to be especially reactive, while the anionic 13-atom cluster exhibits unexpected stability against reactivity with oxygen. Theoretical studies show that the odd even selective behavior is correlated with the excitation needed to activate the O-O bond in O-2. Furthermore, by comparison with the reactivity of proximate even-electron clusters, we demonstrate that the inactivity of Ag-13(-) is associated with its large spin excitation energy, ascribed to a crystal-field-like splitting of the orbitals caused by the bilayer atomic structure, which induces a large gap despite not having a magic number of valence electrons.