We present a Taylor-based theory of deformation of an aggregate of rigid-plastic crystals that allows for heterogeneity of grain deformation, and use it to model macroscopic subdivision of grains into mutually misoriented volumes, a process termed deformation banding. Each grain is assumed to accommodate the macroscopically imposed deformation such that the power of its plastic deformation is minimized. This minimization may involve the formation of deformation bands. The theory is applied to tension, compression and rolling of fcc aluminium and bcc α-iron polycrystals, and used to predict the macroscopic mechanical response, the polycrystal texture, the orientation of deformation bands, and the misorientations across them. These predictions are compared with experimental observations available in the literature, and good qualitative agreement is found.