Using far-ultraviolet (FUV) and near-ultraviolet (NUV) photometry from guest investigator programmes on the Galaxy Evolution Explorer (GALEX) satellite, optical photometry from the MDM Observatory and optical integral-field spectroscopy from SAURON, we explore the UV-line-strength relations of the 48 nearby early-type galaxies in the SAURON sample. Identical apertures are used for all quantities, avoiding aperture mismatch. We show that galaxies with purely old stellar populations show well-defined correlations of the integrated FUV -V and FUV - NUV colours with the integrated Mg b and Hβ absorption line-strength indices, strongest for FUV - NUV. Correlations with the NUV -V colour, Fe5015 index and stellar velocity dispersion σ are much weaker. These correlations put stringent constraints on the origin of the UV-upturn phenomenon in early-type galaxies and highlight its dependence on age and metallicity. In particular, despite recent debate, we recover the negative correlation between FUV -V colour and Mg line strength originally publicized by Burstein et al., which we refer to as the 'Burstein relation', suggesting a positive dependence of the UV upturn on metallicity. We argue that the scatter in the correlations is real and present mild evidence that a strong UV excess is preferentially present in slow-rotating galaxies. We also demonstrate that most outliers in the correlations are galaxies with current or recent star formation, some at very low levels. We believe that this sensitivity to weak star formation, afforded by the deep and varied data available for the SAURON sample, explains why our results are occasionally at odds with other recent but shallower surveys. This is supported by the analysis of a large, carefully crafted sample of more distant early-type galaxies from the Sloan Digital Sky Survey (SDSS), more easily comparable with current and future large surveys.