We have recently found fundamental differences in the radio properties of red quasars when compared to typical blue quasars. In this paper we use data from the X-shooter spectrograph on the Very Large Telescope, providing spectral coverage from ~3000-25000{AA}, of a sample of 40 red and blue luminous quasars at 1.45<z<1.65 to explore the connections between the radio, emission-line, and accretion-disc properties. We fit various dust-extinction curves to the data and find that dust reddening can fully explain the observed colours for the majority of the red quasars in our sample, with moderate extinctions ranging from AV~0.06-0.7mags. We confront our spectra with a simple thin accretion-disc model and find this can describe the continua of both the blue and red quasars, once corrected for dust extinction; we also find no significant differences in the accretion properties. We detect ionized outflows in a number of red and blue quasars, but do not find any significant evidence that they are more prevalent in the red quasar population. Overall our findings imply that the radio emission is more closely connected to circumnuclear/ISM opacity rather than accretion disc or outflow differences.