We present a detailed analysis of Jupiter's X-ray (0.2-10 keV) auroral emissions as observed over two XMM-Newton revolutions in Nov. 2003 and compare it with that of an earlier observation in Apr. 2003. We discover the existence of an electron bremsstrahlung component in the aurorae, which accounts for essentially all the X-ray flux above 2 keV: its presence had been predicted but never detected for lack of sensitivity of previous X-ray missions. This bremsstrahlung component varied significantly in strength and spectral shape over the 3.5 days covered by the Nov. 2003 observation, displaying substantial hardening of the spectrum with increasing flux. This variability may be linked to the strong solar activity taking place at the time, and may be induced by changes in the acceleration mechanisms inside Jupiter's magnetosphere. As in Apr. 2003, the auroral spectra below 2 keV are best fitted by a superposition of line emission most likely originating from ion charge exchange, with OVII playing the dominant role. We still cannot resolve conclusively the ion species responsible for the lowest energy lines (around 0.3 keV), so the question of the origin of the ions (magnetospheric or solar wind) is still open. It is conceivable that both scenarios play a role in what is certainly a very complex planetary structure. High resolution spectra of the whole planet obtained with the XMM-Newton Reflection Grating Spectrometer in the range 0.5-1 keV clearly separate emission lines (mostly of iron) originating at low latitudes on Jupiter from the auroral lines due to oxygen. These are shown to possess very broad wings which imply velocities of ~5000 km s. Such speeds are consistent with the energies at which precipitating and charge exchanging oxygen ions are expected to be accelerated in Jupiter's magnetosphere. Overall we find good agreement between our measurements and the predictions of recently developed models of Jupiter's auroral processes.