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 $(7.1 pm 0.6) 10^{-27} cm^2/H times ( u/353, GHz)^{1.53pm0.03}$ for $100 < u <353$GHz. (2) We map the spectral index of dust emission at millimetre wavelengths, which is remarkably constant at $beta_{mm} = 1.51pm 0.13$. 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 $ u < 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.
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