  We extend our previous work on applying CMB techniques to the mapping of
gravitational-wave backgrounds to backgrounds which have non-GR polarisations.
Our analysis and results are presented in the context of pulsar-timing array
observations, but the overarching methods are general, and can be easily
applied to LIGO or eLISA observations using appropriately modified response
functions. Analytic expressions for the pulsar-timing response to gravitational
waves with non-GR polarisation are given for each mode of a spin-weighted
spherical-harmonic decomposition of the background, which permit the signal to
be mapped across the sky to any desired resolution. We also derive the
pulsar-timing overlap reduction functions for the various non-GR polarisations,
finding analytic forms for anisotropic backgrounds with scalar-transverse
("breathing") and vector-longitudinal polarisations, and a semi-analytic form
for scalar-longitudinal backgrounds. Our results indicate that pulsar-timing
observations will be completely insensitive to scalar-transverse mode
anisotropies in the polarisation amplitude beyond dipole, and anisotropies in
the power beyond quadrupole. Analogously to our previous findings that
pulsar-timing observations lack sensitivity to tensor-curl modes for a
transverse-traceless tensor background, we also find insensitivity to
vector-curl modes for a vector-longitudinal background.
