The latest Gaia Focused Product Release (FPR) provided variability information for ~1000 long-period red giant binaries, including almost ~700 ellipsoidal binary candidates, the largest sample to date of this binary type having both photometric and spectroscopic time series observations. We aim to characterise physically (luminosity, mass, radius) and chemo-dynamically (metallicity, [alpha/Fe], Galactic velocities) the population of long-period red giant ellipsoidal binary candidates, and a subsample of rotational variable candidates, from Gaia FPR, combining Gaia astrometry, photometry and spectroscopy observations. We cross-matched the Gaia DR3 measurements (positions, velocities, atmospheric parameters, chemical abundances) with the catalogue of long-period red giant candidates from the Gaia FPR, having photometric and radial velocity variability information. Combined with the photo-geometric distances, the extinction, bolometric magnitude, luminosity, spectroscopic radius and mass were estimated. The accuracy of this method was tested for similar samples in the literature, including red giant binaries having asteroseismic determination of their physical parameters. Ellipsoidal variables are characterized to be low to intermediate-mass stars (0.6<=M1<=5.0M_{sun}), with radii as large as the Roche lobe radius of the binary. Eccentricities tend to be lower for primary stars with smaller radii, as the expected result of tidal circularization. Combined with the orbital properties, estimates for the minimum mass of the companion agree with the scenario of a low-mass compact object as the secondary star. There are at least 14 ellipsoidal binaries with orbital periods and masses of the two stars compatible with model predictions for Type Ia SN progenitors. For the rotational variables, their orbital periods, enhanced chromospheric activity, smaller radii and low mass (M_1=1.0M_{sun}_, resemble the population of young alpha-rich binaries in the thick disk. An episode of mass transfer in those systems may have produced the enhanced alpha abundances, and the enhanced [Ce/Fe] abundances reported in a few ellipsoidal binaries. Luminosities, radii and masses were derived for 243 ellipsoidal and 39 rotational binary candidates, the largest Galactic sample of these variables, having chemo-dynamical and physical parametrization. Based on their mean chemo-dynamical properties and stellar parameters, these binaries can be considered as two manifestations of the same phenomena, a close binary with a giant primary, instead of two independent, unrelated binary types. Detailed future analysis of individual sources will provide insights into the history and future evolution of these binaries.
Cone search capability for table J/A+A/696/A82/tablee12 (Stellar parameters for the primary star (table E1) and mass estimate, mass ratio, filling factor (table E2))
