Dunes and antidunes on sand and gravel streambeds propagate with a finite speed, with dunes propagating downstream and antidunes propagating both upstream and downstream. Existing linear stability theories predict the dune propagation reasonably well. However, in light of the experimental data, they fail to accurately predict the directional preference of antidunes. We revisit the problem by employing a one-equation rotational-flow model. For bedload-dominant streams, the phase lag between the sediment flux and the local bed topography is the central mechanism causing the instability. We find a clear signature of the non-propagating perturbation within the unstable region of antidunes, separating the upstream and downstream propagating antidunes. The present model offers a unique reconstruction of the linear stability map, capturing satisfactorily the experimental data of dunes, antidunes, and the transition from dunes to antidunes. © 2021 Author(s).