  The search for the primordial B-modes of the cosmic microwave background
(CMB) relies on the separation from the brighter foreground dust signal. In
this context, the characterisation of the spectral energy distribution (SED) of
thermal dust in polarization has become a critical subject of study. We present
a power-spectra analysis of Planck data, which improves on previous studies by
using the newly released SRoll2 maps that correct residual data systematics,
and by extending the analysis to regions near the Galactic plane. Our analysis
focuses on the lowest multipoles between l=4 and 32, and three sky areas with
sky fractions of fsky = 80%, 90%, and 97%. The mean dust SED for polarization
and the 353 GHz Q and U maps are used to compute residual maps at 100, 143 and
217 GHz, highlighting spatial variations of the dust polarization SED.
Residuals are detected at the three frequencies for the three sky areas. We
show that models based on total intensity data are underestimating by a
significant factor the complexity of dust polarized CMB foreground. Our
analysis emphasizes the need to include variations of polarization angles of
the dust polarized CMB foreground. The frequency dependence of the EE and BB
power spectra of the residual maps yields further insight. We find that the
moments expansion to the first order of the modified black-body (MBB) spectrum
provides a good fit to the EE power-spectra. This result suggests that the
residuals could follow mainly from variations of dust MBB spectral parameters.
However, this conclusion is challenged by cross-spectra showing that the
residuals maps at the three frequencies are not fully correlated, and the fact
that the BB power-spectra do not match the first order moment expansion of a
MBB SED. This work sets new requirements for simulations of the dust polarized
foreground and component separation methods (abridged)
