Although we are nearing a consensus that most ULXs are stellar-mass black holes in a super-Eddington state, little is yet established of the physics of this accretion mode. Here, we use a combined X-ray spectral and timing analysis of a sample of ULXs to investigate this new accretion regime. We suggest a spectral classification scheme that separates ULXs into three classes: a broadened disc class, and two-component hard and soft ultraluminous regimes. At the lowest luminosities the ULX population is dominated by sources with broadened disc spectra, whilst two component spectra are seen at higher luminosities, suggestive of a distinction between ~ Eddington and super-Eddington accretion modes. We find high levels of variability are limited to ULXs with soft ultraluminous spectra, and a few broadened disc sources. Furthermore, the variability is strongest at high energies, suggesting it originates in the harder spectral component. These properties are consistent with current models of super-Eddington emission, where a wind forms a funnel around the central regions of the accretion flow. As the wind provides the soft component this suggests that inclination is the key determinant in the observed X-ray spectra, which is very strongly supported by the variability results if this originates due to clumpy material at the edge of the wind intermittently obscuring our line-of-sight to the central regions of the ULX. The pattern of spectral variability with luminosity in two ULXs that straddle the hard/soft ultraluminous regime boundary is consistent with the wind increasing at higher accretion rates, and thus narrowing the opening angle of the funnel. Hence, this work suggests that most ULXs can be explained as stellar-mass black holes accreting at and above the Eddington limit, with their observed characteristics dominated by two variables: accretion rate and inclination. (abridged) ; Comment: Accepted for publication in MNRAS; 20 pages, 6 figures, 4 tables