A star crossing the tidal radius of a supermassive black hole will be spectacularly ripped apart with an accompanying burst of radiation. A few tens of such tidal disruption events have now been identified in optical wavelengths, but the exact origin of the strong optical emission remains inconclusive. Here we report polarimetric observations of three tidal disruption events. The continuum polarization appears independent of wavelength, while emission lines are partially depolarized. These signatures are consistent with photons being scattered and polarized in an envelope of free electrons. An almost axisymmetric photosphere viewed from different angles is in broad agreement with the data, but there is also evidence for deviations from axial symmetry before the peak of the flare and significant time evolution at early times, compatible with the rapid formation of an accretion disk. By combining a super-Eddington accretion model with a radiative transfer code, we simulate the polarization degree as a function of disk mass and viewing angle and we show that the predicted levels are compatible with the observations for extended reprocessing envelopes of similar to 1,000 gravitational radii. Spectropolarimetry therefore constitutes a new observational test for tidal disruption event models, and opens an important new line of exploration in the study of tidal disruption events.