The dust-Hi correlation is used to characterize the emission properties of dust in the diffuse interstellar medium (ISM) from far infrared wavelengths to microwave frequencies. The field of this investigation encompasses the part of the southern sky best suited to study the cosmic infrared and microwave backgrounds. We cross-correlate sky maps from Planck, the Wilkinson Microwave Anisotropy Probe (WMAP), and the diffuse infrared background experiment (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 deg2 centred on the southern Galactic pole. We present a general methodology to study the dust-Hi correlation over the sky, including simulations to quantify uncertainties. Our analysis yields four specific results. (1) We map the temperature, submillimetre emissivity, and opacity of the dust per H-atom. The dust temperature is observed to be anti-correlated with the dust emissivity and opacity. We interpret this result as evidence of dust evolution within the diffuse ISM. The mean dust opacity is measured to be (7.1 ± 0.6) × 10-27 cm2 H-1 × (ν/ 353 GHz)1.53 ± 0.03 for 100 ≤ ν ≤ 353 GHz. This is a reference value to estimate hydrogen column densities from dust emission at submillimetre and millimetre wavelengths. (2) We map the spectral index βmm of dust emission at millimetre wavelengths (defined here as ν ≤ 353 GHz), and find it to be remarkably constant at βmm = 1.51 ± 0.13. We compare it with the far infrared spectral index βFIR derived from greybody fits at higher frequencies, and find a systematic difference, βmm - βFIR = - 0.15, which suggests that the dust spectral energy distribution (SED) flattens at ν ≤ 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 (AME). We show that the flattening of the dust SED can be accounted for with an additional component with a blackbody spectrum. This additional component, which accounts for (26 ± 6)% of the dust emission at 100 GHz, 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 analyse the residuals of the dust-Hi correlation. We identify a Galactic contribution to these residuals, which we model with variations of the dust emissivity on angular scales smaller than that of our correlation analysis. This model of the residuals is used to quantify uncertainties of the CIB power spectrum in a companion Planck paper. Appendices are available in electronic form at http://www.aanda.org