The overall energy or stiffness of an elastic composite depends on the microgeometry. Recently, there has been a lot of work on `extremal microstructures' for elastic composites, for example microstructures which minimize the elastic energy at a given macroscopic strain. However, most attention has been focused on composites made of the elastically isotropic component materials. Breaking with this tradition, we consider composites made of two fully anisotropic phases. Our approach, based on the well-known translation method, provides not only the energy bound but also necessary and sufficient conditions for optimality in terms of the local strain field. These optimality conditions enable us to look for optimal microstructures in a more systematic way than before. They also provide clarification of the relations between different problems, for example bounding effective conductivity of a conducting composite versus minimizing strain energy of an elastic composite. Our analysis shows that anisotropy of the constituent materials is very important in determining optimal microgeometries. Some constructions of extremal matrix-inclusion composites made from isotropic components cease to be available when the matrix material is anisotropic, even when the degree of anisotropy is small. Most of our analysis is restricted to two space dimensions.