  Betelgeuse is an M supergiant that harbors spots and giant granules at its
surface and presents linear polarization of its continuum. We have previously
discovered linear polarization signatures associated with individual lines in
the spectra of cool and evolved stars. Here, we investigate whether a similar
linearly polarized spectrum exists for Betelgeuse. We used the
spectropolarimeter Narval, combining multiple polarimetric sequences to obtain
high signal-to-noise ratio spectra of individual lines, as well as the
least-squares deconvolution (LSD) approach. We have discovered the existence of
a linearly polarized spectrum for Betelgeuse, detecting a rather strong signal
(at a few times 10$^{-4}$ of the continuum intensity level), both in individual
lines and in the LSD profiles. Studying its properties and the signal observed
for the resonant \ion{Na}{i}\,D lines, we conclude that we are mainly observing
depolarization of the continuum by the absorption lines. The linear
polarization of the Betelgeuse continuum is due to the anisotropy of the
radiation field induced by brightness spots at the surface and Rayleigh
scattering in the atmosphere. We have developed a geometrical model to
interpret the observed polarization, from which we infer the presence of two
brightness spots and their positions on the surface of Betelgeuse. We show that
applying the model to each velocity bin along the Stokes Q and U profiles
allows the derivation of a map of the bright spots. We use the Narval linear
polarization observations of Betelgeuse obtained over a period of 1.4 years to
study the evolution of the spots. Our study of the linearly polarized spectrum
of Betelgeuse provides a novel method for studying the evolution of brightness
spots at its surface and complements quasi-simultaneous observations obtained
with PIONIER at the VLTI.
