  The dust-HI correlation is used to characterize the emission properties of
dust in the diffuse interstellar medium. We cross-correlate sky maps from
Planck, WMAP, and DIRBE, at 17 frequencies from 23 to 3000 GHz, with the Parkes
survey of the 21-cm line emission of neutral atomic hydrogen, over a contiguous
area of 7500 deg$^2$ centred on the southern Galactic pole. Our analysis yields
four specific results. (1) The dust temperature is observed to be
anti-correlated with the dust emissivity and opacity. We interpret this result
as evidence for dust evolution within the diffuse ISM. The mean dust opacity is
measured to be $(7.1 \pm 0.6) 10^{-27} cm^2/H \times (\nu/353\,
GHz)^{1.53\pm0.03}$ for $100 < \nu <353$GHz. (2) We map the spectral index of
dust emission at millimetre wavelengths, which is remarkably constant at
$\beta_{mm} = 1.51\pm 0.13$. We compare it with the far infrared spectral index
beta_FIR derived from greybody fits at higher frequencies, and find a
systematic difference, $\beta_{mm}-\beta_{FIR} = -0.15$, which suggests that
the dust SED flattens at $\nu < 353\,$GHz. (3) We present spectral fits of the
microwave emission correlated with HI from 23 to 353 GHz, which separate dust
and anomalous microwave emission. The flattening of the dust SED can be
accounted for with an additional component with a blackbody spectrum, which
accounts for $(26 \pm 6)$% of the dust emission at 100 GHz and could represent
magnetic dipole emission. Alternatively, it could account for an increasing
contribution of carbon dust, or a flattening of the emissivity of amorphous
silicates, at millimetre wavelengths. These interpretations make different
predictions for the dust polarization SED. (4) We identify a Galactic
contribution to the residuals of the dust-HI correlation, which we model with
variations of the dust emissivity on angular scales smaller than that of our
correlation analysis.
