  In the magnetar model, the quiescent non-thermal soft X-ray emission from
Anomalous X-ray Pulsars and Soft-Gamma Repeaters is thought to arise from
resonant comptonization of thermal photons by charges moving in a twisted
magnetosphere. Robust inference of physical quantities from observations is
difficult, because the process depends strongly on geometry and current
understanding of the magnetosphere is not very deep. The polarization of soft
X-ray photons is an independent source of information, and its magnetospheric
imprint remains only partially explored. In this paper we calculate how
resonant cyclotron scattering would modify the observed polarization signal
relative to the surface emission, using a multidimensional Monte Carlo
radiative transfer code that accounts for the gradual coupling of polarization
eigenmodes as photons leave the magnetosphere. We employ a globally-twisted,
self-similar, force-free magnetosphere with a power-law momentum distribution,
assume a blackbody spectrum for the seed photons, account for general
relativistic light deflection close to the star, and assume that vacuum
polarization dominates the dielectric properties of the magnetosphere. The
latter is a good approximation if the pair multiplicity is not much larger than
unity. Phase-averaged polarimetry is able to provide a clear signature of the
magnetospheric reprocessing of thermal photons and to constrain mechanisms
generating the thermal emission. Phase-resolved polarimetry, in addition, can
characterize the spatial extent and magnitude of the magnetospheric twist angle
at ~100 stellar radii, and discern between uni- or bidirectional particle
energy distributions, almost independently of every other parameter in the
system. We discuss prospects for detectability with GEMS.
