Predictions are made based on an analysis of a new nonlinear theory of martensitic transformations introduced by the authors. The crystal is modelled as a nonlinear elastic material, with a free-energy function that is invariant with respect to both rigid-body rotations and the appropriate crystallographic symmetries. The predictions concern primarily the two-well problem, that of determining all possible energy-minimizing deformations that can be obtained with two coherent and macroscopically unstressed variants of martensite. The set of possible macroscopic deformations obtained is completely determined by the lattice parameters of the material. For certain boundary conditions the total free energy does not attain a minimum, and the finer and finer oscillations of minimizing sequences are interpreted as corresponding to microstructure. The predictions are amenable to experimental tests. The proposed tests involve the comparison of the theoretical predictions with the mechanical response of properly oriented plates subject to simple shear. Additional crystallographic background is given for the model, and the theory is compared with the `linearized' model of Khachaturyan, Roitburd and Shatalov. There are some similarities in the predictions of the two theories, but also some major discrepancies.