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A striking magnetization anisotropy in thin films of vanadium-doped stannic oxide is reported and investigated. The single-crystalline Sn1-xVxO2 (0≤x≤0.1) thin films, grown on Al2O3 substrate, are (101) oriented and exhibit a temperature-dependent in-plane anisotropy of the saturation magnetization. The in-plane magnetic moment reaches a maximum close to the [101¯] direction, but the anisotropy axis is incompatible with the crystalline structure of the SnO2 thin films. However, it is consistent with V atoms occupying uniaxially distorted octahedral interstices, thus breaking the symmetry of the film. The moment anisotropy decreases gradually with increasing temperature and persists to temperatures above room temperature. It is modeled as a spin-orbit effect involving the hopping of orbital-current loops, as contrasted to isotropic Heisenberg exchange, and the temperature dependence of the magnetism is explained on the basis of a thermal admixture of nearly degenerate crystal-field states. The phenomena may be relevant to a wide range of dilute magnetic oxides.