The magnetic field and emission-line spectrum of the remarkable white dwarf GD356

New spectropolarimetric observations of the unique white dwarf GD356 are presented. These observations show resolved Zeeman triplets of Hα and Hβ in emission with distinctive circular polarization properties. A detailed modelling of the spectropolarimetric observations points to the existence of a latitudinally extended spherical sector/strip covering approximately 0.1 of the stellar surface over which the stellar atmosphere has an inverted temperature distribution at low optical depths. Magneto-optical effects from regions producing photospheric absorption appear to be responsible for narrow circular polarization features which are observed near the pi component of the emission lines. Radio observations of GD356 were obtained at the Very Large Array at 8439.9 and 4860.1 MHz, to gain clues on the origin of the emission lines, but the failure to detect the star in either frequency band does not provide a useful test for the presence of a corona. Also, new infrared (IR) observations to search for a low-mass stellar companion as a possible source of matter for accretion yielded null results. We have investigated the possibility that the temperature inversion is caused by Bondi-Hoyle accretion from the interstellar medium. The absence of additional components of radiation in the X-ray, ultraviolet, optical and near-IR wavelengths implies that the accretion luminosity must be significantly less than the white dwarf luminosity. The implied low value for the specific accretion rate (m_solar<10^-6 g cm^-2 s^-1) indicates that a shock would not form, but that the atmosphere will be particle heated by direct bombardment by the infalling protons. A detailed model of the bombardment solution in the very low-accretion regime is required before the viability of such a model can be fully evaluated.