  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 Zeeman-induced 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 H$\alpha$, reported previously, are explained by different
radiative anisotropies arising from slabs of different optical depths.
