Polymer blends are formulated by mixing polymers with different chemical structures to create new materials with properties intermediate between those of the individual components. While Flory-Huggins (FH) theory explains some basic trends in blend miscibility, the theory completely neglects the dissimilarity in monomer structures that is central to the fabrication of real blends. We systematically investigate the influence of monomer structure on blend miscibility using a lattice cluster theory (LCT) generalization of the FH model. Analytic calculations are rendered tractable by restricting the theoretical analysis to the limit of incompressible and high molecular weight blends. The well-known miscibility pattern predicted by FH theory is recovered only for a limited range of monomer size and shape asymmetries, but additional contributions to the LCT entropy and internal energy of mixing for polymers with dissimilarly shaped monomers lead to three additional blend miscibilty classes whose behaviors are quite different from the predictions of classical FH theory. One blend miscibility class (class IV) exhibits a remarkable resemblance to the critical behavior of polymer solutions. In particular, the theta temperature for class IV blends is near a molecular weight insensitive critical temperature for phase separation, the critical composition is highly asymmetric, and the correlation length amplitude is significantly less than the chain radius of gyration. Experimental evidence for these new blend miscibility classes is discussed, and predictions are made for specific blends of polyolefins that should illustrate these new patterns of blend miscibility.