Neural networks are currently implemented on digital Von Neumann machines, which do not fully leverage their intrinsic parallelism. We demonstrate how to use a novel class of reconfigurable dynamical systems for analogue information processing, mitigating this problem. Our generic hardware platform for dynamic, analogue computing consists of a reciprocal linear dynamical system with nonlinear feedback. Thanks to reciprocity, a ubiquitous property of many physical phenomena like the propagation of light and sound, the error backpropagation-a crucial step for tuning such systems towards a specific task-can happen in hardware. This can potentially speed up the optimization process significantly, offering important benefits for the scalability of neuro-inspired hardware. In this paper, we show, using one experimentally validated and one conceptual example, that such systems may provide a straightforward mechanism for constructing highly scalable, fully dynamical analogue computers.