The Kirkwood-Salsburg equation for the many-particle distribution function is generalized to mixtures of anisotropic particles which can form ordered (e.g., liquid crystal or solid) phases. We extend similarly various statements of the "zero-separation" theorem. This allows us to obtain the small- and large-" scaled" particle limits in which a single particle is scaled to zero and infinite size, respectively. Simple interpolation between the corresponding solutions to the Kirkwood-Salsburg equation leads directly to the generalized scaled particle theory (SPT). Distribution functions as well as thermodynamic properties are considered. As special cases we treat first the single-component, hard-sphere system in its disordered and ordered states; an additional (gas <=> liquid) phase transition fails to appear when attractions are added. We then consider mixtures of particles, including those which interact via nonadditive hard cores. The SPT relations describing order states and isotropic phases of mixtures of differently shaped particles are found to violate certain of the Maxwell relations. This thermodynamic inconsistency, inherent in the SPT because of its having afforded special status to a single particle, is absent altogether in the alternative description of hard-particle fluids which is presented in an accompanying paper.