  Parity violating extensions of the standard electromagnetic theory cause in
vacuo rotation of the plane of polarization of propagating photons. This
effect, also known as cosmic birefringence, impacts the cosmic microwave
background (CMB) anisotropy angular power spectra, producing non-vanishing
$T$--$B$ and $E$--$B$ correlations that are otherwise null when parity is a
symmetry. Here we present new constraints on an isotropic rotation,
parametrized by the angle $\alpha$, derived from Planck 2015 CMB polarization
data. To increase the robustness of our analyses, we employ two complementary
approaches, in harmonic space and in map space, the latter based on a peak
stacking technique. The two approaches provide estimates for $\alpha$ that are
in agreement within statistical uncertainties and very stable against several
consistency tests. Considering the $T$--$B$ and $E$--$B$ information jointly,
we find $\alpha = 0.31^{\circ} \pm 0.05^{\circ} \, ({\rm stat.})\, \pm
0.28^{\circ} \, ({\rm syst.})$ from the harmonic analysis and $\alpha =
0.35^{\circ} \pm 0.05^{\circ} \, ({\rm stat.})\, \pm 0.28^{\circ} \, ({\rm
syst.})$ from the stacking approach. These constraints are compatible with no
parity violation and are dominated by the systematic uncertainty in the
orientation of Planck's polarization-sensitive bolometers.
