Aims.We study the properties and nature of extremely red galaxies (ERO, R-K?5.6) found behind two lensing clusters and compare them with other known galaxy populations. Methods: New HST/ACS, Spitzer (IRAC and MIPS), and Chandra/ACIS observations of the two lensing clusters Abell 1835 and AC114 have been obtained, combined with our earlier optical and near-IR observations and used to study EROs in these deep fields. Results: We have found 6 and 9 EROs in Abell 1835 and AC114. Several (7) of these objects are undetected up to the I and/or z850 band, and are hence ?optical? drop-out sources (at a 3? limit). The photometric redshifts of most of our sources (80%) are z? 0.7-1.5. According to simple colour-colour diagrams, the majority of our objects would be classified as hosting old stellar populations (?ellipticals?). However, there are clear signs of dusty starbursts for several among them. These objects correspond to the most extreme ones in R-K colour. After correcting for lens magnification, we estimate a surface density of (0.97±0.31) arcmin-2 for EROs with (R-K ? 5.6) at K<20.5. As in earlier studies, an overlap of different populations is found. Among our 15 EROs 6 (40%) also classify as distant red galaxies. Eleven of 13 EROs (85%) with available IRAC photometry also fulfil the selection criteria for IRAC selected EROs (IEROs) of Yan et al. (2004, ApJ, 616, 63). SED modelling shows that ~36% of the IEROs in our sample are luminous or ultra-luminous infrared galaxies ((U)LIRG). Some very red DRGs are found to be very dusty starbursts, even (U)LIRGs. No AGNs is found, although faint activity cannot be excluded for all objects. From mid-IR and X-ray data, 5 objects are clearly classified as starbursts. The derived properties are quite similar to those of DRGs and IEROs, except for 5 extreme objects in terms of colours, for which a very high extinction (AV ? 3) is found. Tables 1-6 are only available in electronic form at http://www.aanda.org Based on observations collected at the Very Large Telescope (Antu/UT1), European Southern Observatory, Paranal, Chile (ESO Programs 69.A-0508, 70.A-0355, 73.A-0471), the NASA/ESA Hubble Space Telescope obtained at the Space Telescope Science Institute which is operated by AURA under NASA contract NAS5-26555, the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under NASA contract 1407, and the Chandra satellite.