The optical/near-infrared (OIR) region of the spectra of low-mass X-ray binaries (XBs) appears to lie at the intersection of a variety of different emission processes. In this paper we present quasi-simultaneous OIR-X-ray observations of 33 XBs in an attempt to estimate the contributions of various emission processes in these sources, as a function of X-ray state and luminosity. A global correlation is found between OIR and X-ray luminosity for low-mass black hole candidate XBs (BHXBs) in the hard X-ray state, of the form LOIR ~ L0.6X. This correlation holds over eight orders of magnitude in LX and includes data from BHXBs in quiescence and at large distances [Large Magellanic Cloud (LMC) and M31]. A similar correlation is found in low-mass neutron star XBs (NSXBs) in the hard state. For BHXBs in the soft state, all the near-infrared (NIR) and some of the optical emissions are suppressed below the correlation, a behaviour indicative of the jet switching off/on in transition to/from the soft state. We compare these relations to theoretical models of a number of emission processes. We find that X-ray reprocessing in the disc and emission from the jets both predict a slope close to 0.6 for BHXBs, and both contribute to the OIR in BHXBs in the hard state, the jets producing ~90 per cent of the NIR emission at high luminosities. X-ray reprocessing dominates the OIR in NSXBs in the hard state, with possible contributions from the jets (only at high luminosity) and the viscously heated disc. We also show that the optically thick jet spectrum of BHXBs extends to near the K band. OIR spectral energy distributions of 15 BHXBs help us to confirm these interpretations. We present a prediction of the LOIR-LX behaviour of a BHXB outburst that enters the soft state, where the peak LOIR in the hard state rise is greater than in the hard state decline (the well-known hysteretical behaviour). In addition, it is possible to estimate the X-ray, OIR and radio luminosity and the mass accretion rate in the hard state quasi-simultaneously, from observations of just one of these wavebands, since they are all linked through correlations. Finally, we have discovered that the nature of the compact object, the mass of the companion and the distance/reddening can be constrained by quasi-simultaneous OIR and X-ray luminosities.