In a previous paper, we have constructed a family of self-consistent triaxial models of quasi-relaxed stellar systems, shaped by the tidal field of the hosting galaxy, as an extension of the well-known spherical King models. For a given tidal field, the models are characterized by two physical scales (such as total mass and central velocity dispersion) and two dimensionless parameters (the concentration parameter and the tidal strength). The most significant departure from spherical symmetry occurs when the truncation radius of the corresponding spherical King model is of the order of the tidal radius, which, for a given tidal strength, is set by the maximum concentration value admitted. For such maximally extended (or "critical") models, the outer boundary has a generally triaxial shape, given by the zero-velocity surface of the relevant Jacobi integral, which is basically independent of the concentration parameter. In turn, the external tidal field can give rise to significant global departures from spherical symmetry (as measured, for example, by the quadrupole of the mass distribution of the stellar system) only for low-concentration models, for which the allowed maximal value of the tidal strength can be relatively high. In this paper, we describe in systematic detail the intrinsic and the projected structure and kinematics of the models, covering the entire parameter space, from the case of sub-critical (characterized by "underfilling" of the relevant Roche volume) to that of critical models. The intrinsic properties can be a useful starting point for numerical simulations and other investigations that require initialization of a stellar system in dynamical equilibrium. The projected properties are a key step in the direction of a comparison with observed globular clusters and other candidate stellar systems.