  The polarized Sunyaev-Zel'dovich (pSZ) effect is sourced by the Thomson
scattering of CMB photons from distant free electrons and yields a novel view
of the CMB quadrupole throughout the observable Universe. Galaxy shear measures
the shape distortions of galaxies, probing both their local environment and the
intervening matter distribution. Both observables give interesting constraints
on the cosmological model; here we ask: what can be learnt from their
combination? The pSZ-shear cross-spectrum measures the
shear-galaxy-polarization bispectrum and contains contributions from (1) the
Sachs-Wolfe (SW) effect, (2) the integrated Sachs-Wolfe (ISW) effect, and (3)
inflationary gravitational waves. Since the modes contributing to the pSZ
signal are not restricted to the Earth's past lightcone, the low-redshift
cross-spectra could provide a novel constraint on dark energy via the ISW
effect, whilst the SW signal couples scalar modes at very different times but
at similar positions; this provides a unique probe of the Universe's
homogeneous time evolution. We give expressions for all major contributions to
the shear, galaxy, and pSZ auto- and cross-spectra, and evaluate their
detectability via Fisher forecasts. Despite significant theoretical utility,
the cross-spectra will be challenging to detect: combining CMB-S4 with Rubin
yields a $1.6\sigma$ detection of the ISW contribution, which increases to
$5.2\sigma$ for a futuristic experiment involving CMB-HD and a higher galaxy
sample density. For parity-even (parity-odd) tensors, we predict a $1\sigma$
limit of $\sigma(r) = 0.9$ ($0.2$) for CMB-S4 and Rubin, or $0.3$ ($0.06$) for
the more futuristic setup. Whilst this is significantly better than the
constraints from galaxy shear alone (and less sensitive to systematics), it is
unlikely to be competitive, but may serve as a useful cross-check.
