We present a detailed investigation of the relation between the orbital content of merger remnants and observable properties of elliptical and S0 galaxies. Our analysis is based on the statistical sample of collisionless mergers of disc galaxies with different mass ratios and orbital parameters, published by Naab & Burkert. We use the spectral method by Carpintero & Aguilar to determine the orbital content of every remnant and correlate it with its intrinsic shape, and its projected kinematic and photometric properties. We discuss the influence of the bulge component and varying pericentre distances. The classified orbit families are box orbits, minor-axis tubes, inner and outer major-axis tubes and boxlets. In general, box orbits dominate the inner parts of the remnant. Major and minor-axis tubes become dominant at intermediate radii and boxlets at large radii. The two most abundant orbit classes are the minor-axis tubes and the box orbits. Their ratio seems to determine the basic properties of a remnant. On average, the fraction of minor-axis tubes increases by a factor of 2 from a merger mass ratio of 1:1 to 4:1, whereas the fraction of box orbits decreases by 10 per cent. At a given mass the central velocity dispersion of a remnant scales with the ratio of minor-axis tubes to box orbits. Interestingly, the division line between rotational supported systems and pressure supported objects, (υmaj/σ0)*= 0.7, turns out to coincide with a box to minor-axis tube ratio of unity. The observed h3-υ/σ anticorrelation for ellipticals cannot be reproduced by collisionless merger remnants. We propose that this can only be reconciled by an additional physical process that significantly reduces the box orbit content. Remnants that are dominated by minor-axis tube orbits have predominantly discy projections. Boxy remnants have always a box to minor-axis tube ratio larger than one. This study will enable us to identify observed ellipticals that could have formed, in the collisionless limit, by gas-poor disc mergers. In addition, it demonstrates how observable properties of spheroidal stellar systems are connected with their intrinsic orbital structure.