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Lithium can be reversibly intercalated into layered Li1+xV1-xO2 (LiCoO2 structure) at 0.1 V, but only if x0. The low voltage combined with a higher density than graphite results in a higher theoretical volumetric energy density; important for future applications in portable electronics and electric vehicles. Here we investigate the crucial question, why Li cannot intercalate into LiVO2 but Li-rich compositions switch on intercalation at an unprecedented low voltage for an oxide? We show that Li+ intercalated into tetrahedral sites are energetically more stable for Li-rich compositions, as they share a face with Li+ on the V site in the transition metal layers. Li incorporation triggers shearing of the oxide layers from cubic to hexagonal packing because the Li2VO2 structure can accommodate two Li per formula unit in tetrahedral sites without face sharing. Such understanding is important for the future design and optimization of low-voltage intercalation anodes for lithium batteries.