The entropy driven demixing in binary mixtures of rigidly ordered hard particles is studied herein. To that end, a free energy functional for this type of system, as well as an approximation for the corresponding pair direct correlation function, are proposed. The accuracy of the functional is first tested by computing the pressure for previously studied similar systems. The spinodal of the fluid-fluid separation is studied, and shown to be a loop in the (rho(1),rho(2)) density-density phase diagram, with both a lower and an upper critical point. This feature is strikingly different from what is observed in the demixing of disordered mixtures, where only a lower critical point is generally found. The influence of molecular shape is equally studied, by varying geometric properties such as convexity, aspect and breadth ratios as well as by mixing prolate and oblate shapes. Within the present theory, demixing is forbidden for mixtures of hard spheres and mixtures of prolate (oblate) shapes with the same breadth, irrespective of their aspect ratios. Demixing is enhanced for mixtures of prolates and oblates and more generally for large asymmetry in the particles breadths. The proposed functional is built such that it reduces to the correct Onsager limit when very large aspect ratios are considered. However, the Onsager second virial approximation, when applied to cases of moderate to small aspect ratios, differs from the present theory, as it can only exhibit demixing with a lower critical point.