  We present a formalism that incorporates hydrogen Lyman-alpha (Ly$\alpha$)
polarization arising from the scattering of radiation in galaxy halos into the
intensity mapping approach. Using the halo model, and emission profiles based
on simulations and observations, we calculate auto and cross power spectra at
redshifts $3\leq z \leq13$ for the Ly$\alpha$ total intensity, $I$, polarized
intensity, $P$, degree of polarization, $\Pi=P/I$, and two new quantities, the
astrophysical $E$ and $B$ modes of Ly$\alpha$ polarization. The one-halo terms
of the $\Pi$ power spectra show a turnover that signals the average extent of
the polarization signal, and thus the extent of the scattering medium. The
position of this feature depends on redshift, and on the specific emission
profile shape and extent, in our formalism. Therefore, the comparison of
various Ly$\alpha$ polarization quantities and redshifts can break degeneracies
between competing effects, and it can reveal the true shape of the emission
profiles, and thus, the physical properties of the cool gas in halos.
Furthermore, measurements of Ly$\alpha$ $E$ and $B$ modes may be used as probes
of galaxy evolution, because they are related to the average degree of
anisotropy in the emission and in the halo gas distribution across redshifts.
The detection of the polarization signal at $z \sim 3-5$ requires improvements
in the sensitivity of current ground-based experiments by a factor of $\sim
10$, and of $\sim 100$ for space-based instruments targeting the redshifts
$z\sim 9-10$, the exact values depending on the specific redshift and
experiment. Interloper contamination in polarization is expected to be small,
because the interlopers need to also be polarized. Overall, Ly$\alpha$
polarization boosts the amount of physical information retrievable on galaxies
and their surroundings, most of it not achievable with total emission alone.
(abridged)
