A striking feature of many natural dynamos is their ability to undergo polarity reversals(1,2). The best documented example is Earth's magnetic field, which has reversed hundreds of times during its history(3,4). The origin of geomagnetic polarity reversals lies in a magnetohydrodynamic process that takes place in Earth's core, but the precise mechanism is debated(5). The majority of numerical geodynamo simulations that exhibit reversals operate in a regime in which the viscosity of the fluid remains important, and in which the dynamo mechanism primarily involves stretching and twisting of field lines by columnar convection(6). Here we present an example of another class of reversing-geodynamo model, which operates in a regime of comparatively low viscosity and high magnetic diffusivity. This class does not fit into the paradigm of reversal regimes that are dictated by the value of the local Rossby number (the ratio of advection to Coriolis force)(7,8). Instead, stretching of the magnetic field by a strong shear in the east-west flow near the imaginary cylinder just touching the inner core and parallel to the axis of rotation is crucial to the reversal mechanism in our models, which involves a process akin to kinematic dynamo waves(9,10). Because our results are relevant in a regime of low viscosity and high magnetic diffusivity, and with geophysically appropriate boundary conditions, this form of dynamo wave may also be involved in geomagnetic reversals.