We search the literature for reports on the spectral properties of neutron star low-mass X-ray binaries when they have accretion luminosities between 1034 and 1036 erg s-1, corresponding to roughly 0.01-1 per cent of the Eddington accretion rate for a neutron star. We found that in this luminosity range the photon index (obtained from fitting a simple absorbed power law in the 0.5-10 keV range) increases with decreasing 0.5-10 keV X-ray luminosity (i.e. the spectrum softens). Such behaviour has been reported before for individual sources, but here we demonstrate that very likely most (if not all) neutron star systems behave in a similar manner and possibly even follow a universal relation. When comparing the neutron star systems with black hole systems, it is clear that most black hole binaries have significantly harder spectra at luminosities of 1034-1035 erg s-1. Despite a limited number of data points, there are indications that these spectral differences also extend to the 1035-1036 erg s-1 range, but above a luminosity of 1035 erg s-1 the separation between neutron star and black hole systems is not as clear as below. In addition, the black hole spectra only become softer below luminosities of 1034 erg s-1 compared to 1036 erg s-1 for the neutron star systems. This observed difference between the neutron star binaries and black hole ones suggests that the spectral properties (between 0.5 and 10 keV) at 1034-1035 erg s-1 can be used to tentatively determine the nature of the accretor in unclassified X-ray binaries. More observations in this luminosity range are needed to determine how robust this diagnostic tool is and whether or not there are (many) systems that do not follow the general trend. We discuss our results in the context of properties of the accretion flow at low luminosities and we suggest that the observed spectral differences likely arise from the neutron star surface becoming dominantly visible in the X-ray spectra. We also suggest that both the thermal component and the non-thermal component might be caused by low-level accretion on to the neutron star surface for luminosities below a few times 1034 erg s-1.