  Full-sky CMB maps from the 2015 Planck release allow us to detect departures
from global isotropy on the largest scales. We present the first searches using
CMB polarization for correlations induced by a non-trivial topology with a
fundamental domain intersecting, or nearly intersecting, the last scattering
surface (at comoving distance $\chi_{rec}$). We specialize to flat spaces with
toroidal and slab topologies, finding that explicit searches for the latter are
sensitive to other topologies with antipodal symmetry. These searches yield no
detection of a compact topology at a scale below the diameter of the last
scattering surface. The limits on the radius $R_i$ of the largest sphere
inscribed in the topological domain (at log-likelihood-ratio $\Delta\ln{L}>-5$
relative to a simply-connected flat Planck best-fit model) are
$R_i>0.97\chi_{rec}$ for the cubic torus and $R_i>0.56\chi_{rec}$ for the slab.
The limit for the cubic torus from the matched-circles search is numerically
equivalent, $R_i>0.97\chi_{rec}$ (99% CL) from polarisation data alone. We also
perform a Bayesian search for a Bianchi VII$_h$ geometry. In the non-physical
setting where the Bianchi cosmology is decoupled from the standard cosmology,
Planck temperature data favour the inclusion of a Bianchi component. However,
the cosmological parameters generating this pattern are in strong disagreement
with those found from CMB anisotropy data alone. Fitting the induced
polarization pattern for this model to Planck data requires an amplitude of
$-0.1\pm0.04$ compared to +1 if the model were to be correct. In the physical
setting where the Bianchi parameters are fit simultaneously with the standard
cosmological parameters, we find no evidence for a Bianchi VII$_h$ cosmology
and constrain the vorticity of such models to $(\omega/H)_0<7.6\times10^{-10}$
(95% CL). [Abridged]
