We use a compilation of disc galaxy rotation curves to assess the role of the luminous component ('baryons') in the rotation curve diversity problem. As in earlier work, we find that rotation curve shape correlates with baryonic surface density: high surface density galaxies have rapidly rising rotation curves consistent with cuspy cold dark matter haloes; slowly rising rotation curves (characteristic of galaxies with inner mass deficits or 'cores') occur only in low surface density galaxies. The correlation, however, seems too weak to be the main driver of the diversity. In addition, dwarf galaxies exhibit a clear trend, from 'cuspy' systems where baryons are unimportant in the inner mass budget to 'cored' galaxies where baryons actually dominate. This trend constrains the various scenarios proposed to explain the diversity, such as (i) baryonic inflows and outflows during galaxy formation; (ii) dark matter self-interactions; (iii) variations in the baryonic mass structure coupled to rotation velocities through the 'mass discrepancy-acceleration relation' (MDAR); or (iv) non-circular motions in gaseous discs. Together with analytical modelling and cosmological hydrodynamical simulations, our analysis shows that each of these scenarios has promising features, but none seems to fully account for the observed diversity. The MDAR, in particular, is inconsistent with the observed trend between rotation curve shape and baryonic importance; either the trend is caused by systematic errors in the data or the MDAR does not apply. The origin of the dwarf galaxy rotation curve diversity and its relation to the structure of cold dark matter haloes remains an open issue.