RT Journal Article T1 Stars and brown dwarfs in the sigma Orionis cluster IV. IDS/INT and OSIRIS/GTC spectroscopy and Gaia DR2 astrometry A1 Caballero, J. A. A1 Burgos, A. de A1 Alonso Floriano, F.J. A1 Cabrera Lavers, A. A1 García Álvarez, D. A1 Montes Gutiérrez, David AB Context. Only a few open clusters are as important for the study of stellar and substellar objects, and their formation and evolution, as the young σ Orionis cluster. However, a complete spectroscopic characterisation of its whole stellar population is still missing. Aims. We filled most of that gap with a large spectroscopic and astrometric survey of targets towards σ Orionis. Eventually, it will be one of the open clusters with the lowest proportion of interlopers and the largest proportion of confirmed cluster members with known uncontrovertible youth features. Methods. We acquired 317 low-resolution optical spectra with the Intermediate Dispersion Spectrograph (IDS) at the 2.5 m Isaac Newton Telescope (INT) and the Optical System for Imaging and low Resolution Integrated Spectroscopy (OSIRIS) at the 10.4 m Gran Telescopio Canarias (GTC). We measured equivalent widths of Li i, Hα, and other key lines from these spectra, and determined spectral types. We complemented this information with Gaia DR2 astrometric data and other features of youth (mid-infrared excess, X-ray emission) compiled with Virtual Observatory tools and from the literature. Results. Of the 168 observed targets, we determined for the first time spectral types of 39 stars and equivalent widths of Li i and Hα of 34 and 12 stars, respectively. We identified 11 close (ρ