Why are sensory signals and motor command signals combined in the neurons of origin of the spinocerebellar pathways and why are the granule cells that receive this input thresholded with respect to their spike output? In this paper, we synthesize a number of findings into a new hypothesis for how the spinocerebellar systems and the cerebellar cortex can interact to support coordination of our multi-segmented limbs and bodies. A central idea is that recombination of the signals available to the spinocerebellar neurons can be used to approximate a wide array of functions including the spatial and temporal dependencies between limb segments, i.e. information that is necessary in order to achieve coordination. We find that random recombination of sensory and motor signals is not a good strategy since, surprisingly, the number of granule cells severely limits the number of recombinations that can be represented within the cerebellum. Instead, we propose that the spinal circuitry provides useful recombinations, which can be described as linear projections through aspects of the multi-dimensional sensorimotor input space. Granule cells, potentially with the aid of differentiated thresholding from Golgi cells, enhance the utility of these projections by allowing the Purkinje cell to establish piecewise-linear approximations of non-linear functions. Our hypothesis provides a novel view on the function of the spinal circuitry and cerebellar granule layer, illustrating how the coordinating functions of the cerebellum can be crucially supported by the recombinations performed by the neurons of the spinocerebellar systems.