We present accurate radial velocities and photoelectric UBV photometry for 73 and 57 red-giant candidates, respectively, in the intermediate-age open clusters NGC 2324, 2818, 3960 and 6259. These data confirm the membership of 47 stars, 12 of which (26%) are spectroscopic binaries; three preliminary orbits have been determined in NGC 3960. From Washington photometry of 8 red giant members, the metallicity of NGC 6259 is found to be [Fe/H] = 0.06 +/- 0.08. At the age of these clusters, most of the red giants are observed in the core-helium (clump) burning phase, the general morphology of which is well reproduced by theoretical models with convective overshooting. However, a number of bona fide cluster giant members are found significantly to the red of the isochrones fitting the rest of the CMD of these and a few other clusters. Some of these stars are binaries, but others seem to be single. In either case, their red colours and/or low luminosities remain unexplained by current stellar evolution theory. Based on observations collected with the Danish 1.54-m telescope at the European Southern Observatory, La Silla (Chile) and at Cerro Tololo Inter-American Observatory, National Optical Astronomy Observatories, operated by the Association of the Universities for Research in Astronomy, Inc., under contract with the National Science Foundation.