After a review of a compact formulation of degenerate and quasidegenerate perturbation theory and the related effective Hamiltonians, a new analysis of the Foldy-Wouthuysen (FW) transformation as a sequence of two transformations is given, where the first transformation decouples electronic from positronic states, while the second transformation reestablishes the normalization. It is then shown that other transformations are possible which share the first part with the FW transformation, but which differ in the second part, by choosing different model spaces. The notorious singularities of the FW transformation in the presence of a Coulomb potential are entirely due to the second step, where the model space is that of all electronic states. Direct perturbation theory as well as its quasidegenerate generalization result as special cases with model spaces related to eigenfunctions of the nonrelativistic Hamiltonian. Additional flexibility results from the option between a unitary transformation and a transformation in intermediate normalization. Nonperturbative alternatives to the perturbation expansion are discussed. The new formalism is particularly useful for the application of direct perturbation theory to relativistic multiconfiguration SCF theory.