  Planck has mapped the intensity and polarization of the sky at microwave
frequencies with unprecedented sensitivity. We make use of the Planck 353 GHz
I, Q, and U Stokes maps as dust templates, and cross-correlate them with the
Planck and WMAP data at 12 frequencies from 23 to 353 GHz, over circular
patches with 10 degree radius. The cross-correlation analysis is performed for
both intensity and polarization data in a consistent manner. We use a mask that
focuses our analysis on the diffuse interstellar medium at intermediate
Galactic latitudes. We determine the spectral indices of dust emission in
intensity and polarization between 100 and 353 GHz, for each sky-patch. The
mean values, $1.59\pm0.02$ for polarization and $1.51\pm0.01$ for intensity,
for a mean dust temperature of 19.6 K, are close, but significantly different
($3.6\,\sigma$). We determine the mean spectral energy distribution (SED) of
the microwave emission, correlated with the 353 GHz dust templates, by
averaging the results of the correlation over all sky-patches. We find that the
mean SED increases for decreasing frequencies at $\nu < 60$ GHz, for both
intensity and polarization. The rise of the polarization SED towards low
frequencies may be accounted for by a synchrotron component correlated with
dust, with no need for any polarization of the anomalous microwave emission. We
use a spectral model to separate the synchrotron and dust polarization and to
characterize the spectral dependence of the dust polarization fraction. The
polarization fraction ($p$) of the dust emission decreases by $(21\pm6)$ % from
353 to 70 GHz. The decrease of $p$ could indicate differences in polarization
efficiency among components of interstellar dust (e.g., carbon versus silicate
grains). Our observational results provide inputs to quantify and optimize the
separation between Galactic and cosmological polarization.
