Recent numerical simulations of two-dimensional convection (compressible and Boussinesq) in the presence of a vertical magnetic field reveal that in some circumstances, narrow rolls are unstable to horizontal shear: tilted rolls are observed, as well as oscillating shearing motion. During the oscillation, the rolls tilt over and are replaced by a vigorous horizontal streaming motion, which decays, and the rolls are reformed, only to tilt over again, either in the same or in the opposite direction. A low-order model of this problem is constructed by truncating the PDEs for Boussinesq magnetoconvection. In the model, oscillatory shearing motion is created either in a ℍ bifurcation from untilted rolls, in which case the rolls tilt first one way and then the other, or in a ℍ bifurcation from tilted rolls, in which case the rolls always tilt in the same direction. Oscillations of the second type are converted into oscillations of the first type in a gluing bifurcation. This scenario is interpreted in terms of a Takens–Bogdanov bifurcation.