Across many brain areas, neurons produce heterogeneous, seemingly disordered responses. Yet such circuits cannot be purely random, since they must possess some structure to generate the representations and computations underlying behavior. How much structure is present in recurrent connectivity relative to disorder, and how the interaction between the two shapes population dynamics and single-neuron responses, remain incompletely understood. Recurrent neural networks trained to perform tasks have become a leading model of such circuits, but conventional training yields a single point in a vast space of task-compatible solutions, with no systematic way to explore this space and no theory of how internal representations vary within it. Without such a theory, the questions above cannot be addressed, and comparisons between trained networks and neural data are difficult to interpret. Here, we introduce a control parameter that governs the degree to which learning reshapes recurrent connectivity, interpolating between a reservoir regime and one in which recurrent weights are restructured by learning to produce task-relevant internal representations. Varying this parameter generates a family of task-compatible solutions whose internal dynamics differ in a controlled and interpretable way. We derive a dynamical mean-field theory showing that, while population-level dynamics converge to a deterministic limit, individual neurons are driven by independent samples from a single-neuron input-current distribution. When connectivity is random, this distribution is Gaussian. Recurrent restructuring drives it toward task-dependent, non-Gaussian forms. In linear networks, restructuring amplifies task-relevant frequencies. In nonlinear networks, it drives a phase transition from chaotic, high-dimensional activity to ordered, low-dimensional dynamics that generalize temporally beyond the training period. We apply the theory to a reaching task in which a recurrent network must reproduce macaque muscle activity, and find that optimally matching simultaneous motor-cortex recordings requires only a small degree of restructuring, with learned structure coexisting with random heterogeneity. These results suggest a broader picture in which large recurrent circuits are largely random but contain, to varying degrees, structured recurrent connectivity sufficient for generalizable, task-relevant representations.