The paper considers the thermodynamic and phase ordering properties of a multi-component Zwanzig mixture of hard rectangular biaxial parallelepipeds. An equation of state (EOS) is derived based on an estimate of the number of arrangements of the particles on a three- dimensional cubic lattice. The methodology is a generalization of the Flory-DiMarzio counting scheme, but, unlike previous work, this treatment is thermodynamically consistent. The results are independent of the order in which particles are placed on the lattice. By taking the limit of zero lattice spacing, a translationally continuous variant of the model (the off-lattice variant) is obtained. The EOS is identical to that obtained previously by a wide variety of different approaches. In the off-lattice limit, it corresponds to a third-level y-expansion and, in the case of a binary mixture of square platelets, it also corresponds to the EOS obtained from fundamental measure theory. On the lattice it is identical to the EOS obtained by retaining only complete stars in the virial expansion. The off-lattice theory is used to study binary mixtures of rods (R₁∈-∈R₂) and binary mixtures of platelets (P₁∈-∈P₂). The particles were uniaxial, of length (thickness) L and width D. The aspect ratios Γ_i∈= ∈L_i/D_i of the components were kept constant (Γ_1R∈=∈15, Γ_1P∈=∈1/15 and Γ_2R∈=∈150, Γ_2P∈=∈1/150), so the second virial coefficient of R₁ was identical to P₁ and similarly for R₂ and P₂. The volume ratio of particles 1 and 2, v₁/v₂, was then varied, with the constraints that v_iR∈= ∈v_iP and Results on nematic-isotropic (N∈-∈I) phase coexistence at an infinite dilution of component 2, are qualitatively similar for rods and platelets. At small values of the ratio v₁/v₂, the addition of component 2 (i.e. a thin rod (e.g. a polymer) or a thin plate) results in the stabilization of the nematic phase. For larger values of v₁/v₂, however, this effect is reversed and the addition of component 2 destabilizes the nematic. For similar molecular volumes of the two components strong fractionation is observed: shorter rods and thicker platelets congregate in the isotropic phase. In general, the stabilization of the ordered phase and the fractionation between the phases are both weaker in the platelet mixtures. The calculated spinodal curves for isotropic-isotropic demixing are noticeably different between the R₁∈-∈R₂ and the P₁∈-∈P₂ systems. The platelet mixtures turn out to be stable with respect to de-mixing up to extremely high densities. The values of the consolute points for the R₁∈-∈R₂ blends are remarkably similar to those obtained using the Parsons-Lee approximation for bi-disperse mixtures of freely rotating cylinders with similar aspect ratios [S. Varga. A. Galindo, G. Jackson, Mol. Phys., 101, 817 (2003)]. In a number of R₁∈-∈R₂ mixtures, phase diagrams exhibiting both N∈-∈I equilibrium and I∈-∈I de-mixing were calculated. The latter is pre-empted by nematic ordering in all the cases studied. Calculations show the possible appearance of azeotropes in the N∈-∈I coexistence domain.