Reverse routing can be used to transfer flood- or pollution-related information monitored at a downstream gauging station to an ungauged upstream cross-section. This signal identification problem is ill-posed and, as such, is sensitive to perturbations in the data to be inverted; therefore, the amplification of errors, e.g., those befalling measurements, must be controlled. Storage routing models are parsimonious diffusion wave substitutes and well suited for conversion to direct reverse routers. We present efficient inversion frameworks based on the lag-and-route (single reservoir plus exact reverse lag-step) and the reservoirs-in-series models. In both cases we invert a centred finite difference scheme of the reservoir storage balance equation that involves only one value of the unknown signal; signal values identified in previous reverse time steps, which would carry perturbations, are absent. This simple structure endows the reverse scheme with robustness. Procedures are verified with perfect and with error-seeded data; solution oscillations caused by the latter are damped by low-pass filtering. Both inverse routing models regain the upstream signals with high fidelity. Reverse storage routing is exemplified in a demonstration of reservoir control and in a field case of solute transport in a stream.