  Pulsar-timing collaborations have recently reported evidence for the
detection of an isotropic stochastic gravitational-wave background consistent
with one sourced by a population of inspiralling supermassive black hole
binaries. However, a certain degree of anisotropy and polarization may be
present. Thus, the characterization of the energy density and polarization of
the background at different angular scales is important. In this paper, we
describe the signatures of linear polarization in the stochastic
gravitational-wave background on the timing residuals obtained with
pulsar-timing arrays. We expand the linear polarization map in terms of
spin-weighted spherical harmonics and recast it into the $E$-mode (parity even)
and $B$-mode (parity odd) basis. We provide expressions for the
minimum-variance estimators for the coefficients of that expansion and evaluate
the smallest detectable signal as a function of the signal-to-noise ratio with
which the isotropic GW signal is detected and the number of pulsars in the
survey. We evaluate the covariance between the estimators for the
spherical-harmonic coefficients of the linear polarization $E$-modes and those
for the intensity anisotropy. We also show that there is no covariance between
the spherical-harmonic coefficients for the $B$-modes of the linear
polarization and those for the circular polarization, even though both have the
same parity. Our approach results in simple, elegant, and easily evaluated
expressions for the overlap reduction functions for linear polarization.
