We analyze spectropolarimetric data of the He I 1083 nm multiplet (1s2s 3 S 1 − 1s2p 3 P o 2,1,0 ) during the X1 flare SOL2014-03-29T17:48, obtained with the Facility Infrared Spectrometer (FIRS) at the Dunn Solar Telescope. While scanning active region NOAA 12017, the FIRS slit crossed a flare ribbon during the impulsive phase, when the helium line intensities turned into emission at ∼ < twice the continuum intensity. Their linear polarization profiles are of the same sign across the multiplet including 1082.9 nm, intensity-like, at ∼ < 5% of the continuum intensity. Weaker Zeemaninduced linear polarization is also observed. Only the strongest linear polarization coincides with hard X-ray (HXR) emission at 30-70 keV observed by the Reuven Ramaty High Energy Solar Spectroscope Imager. The polarization is generally more extended and lasts longer than the HXR emission. The upper J = 0 level of the 1082.9 nm component is unpolarizable, thus lower level polarization is the culprit. We make non-LTE radiative transfer calculations in thermal slabs optimized to fit only intensities. The linear polarizations are naturally reproduced, through a systematic change of sign with wavelength of the radiation anisotropy when slab optical depths of the 1082.9 component are ∼ < 1. Collisions with beams of particles are neither needed nor can they produce the same sign of polarization of the 1082.9 and 1083.0 nm components. The He I line polarization merely requires heating sufficient to produce slabs of the required thickness. Widely different polarizations of Hα, reported previously, are explained by different radiative anisotropies arising from slabs of different optical depths.