We study the relations between the X-ray luminosity, orbital period and absolute near-infrared magnitude of persistent low-mass X-ray binaries (LMXBs). We show that often the optical and near-infrared spectral energy distributions of LMXBs can be adequately described by a simple model of an accretion disc and a secondary star reprocessing X-ray emission from a central compact object. This gives us evidence that one can make a reliable estimate of the orbital period of a persistent LMXB using an X-ray luminosity and an absolute infrared magnitude. Using a sample of well-known LMXBs, we have constructed a correlation of LX, Porb and MK values that can be approximated by a straight line with root-mean-square scatter at the level of ∼0.3 mag. Such a correlation, being to some extent analogous to the correlation found by van Paradijs & McClintock, might be helpful for future population studies, especially in the light of forthcoming surveys of the Galaxy in the X-ray and infrared spectral domains.