  (Abridged) Polarized emission from masers is an excellent tool to study
magnetic fields in maser sources. The linear polarization of most masers is
understood as an interplay of maser saturation and anisotropic pumping.
However, for the latter mechanism, no quantitative modeling has been presented
yet. We present a comprehensive model of maser polarization, including
quantitative modeling of both anisotropic pumping and the effects of maser
saturation on the polarization of masers. We extend regular maser excitation
modeling with a dimension that describes the molecular population alignments,
as well as including the linear polarization dimension to the radiative
transfer. The results of the excitation analysis yield the anisotropic pumping
and decay parameters, that are subsequently used in one-dimensional proper
maser polarization radiative transfer modeling. We present the anisotropic
pumping parameters for a variety of transitions from class I CH$_3$OH masers,
H$_2$O masers and SiO masers. SiO masers are highly anisotropically pumped due
to them occurring in the vicinity of a late-type star, that irradiates the
maser region with a strong directional radiation field. Class I CH$_3$OH masers
and H$_2$O masers occur in association with shocks, and they are modestly
anisotropically pumped due to the anisotropy of the excitation region. Our
modeling constitutes the first quantitative constraints on the anisotropic
pumping of masers. We find that anisotropic pumping can explain the high
polarization yields of SiO masers, as well as the modest polarization of
unsaturated class I CH$_3$OH masers. We predict that the $183$ GHz H$_2$O maser
is strongly anisotropically pumped. Finally, we outline a mechanism through
which non-Zeeman circular polarization is produced, when the magnetic field
changes direction along the propagation through an anisotropically pumped
maser.
