Stably stratified atmospheric boundary layers are often characterized by a veering wind profile, in which the wind direction changes clockwise with height in the Northern Hemisphere. Wind-turbine wakes respond to this veer in the incoming wind by stretching from a circular shape into an ellipsoid. We investigate the relationship between this stretching and the direction of the turbine rotation by means of large-eddy simulations. Clockwise rotating, counterclockwise rotating, and non-rotating actuator disc turbines are embedded in wind fields of a precursor simulation with no wind veer and in wind fields with a Northern Hemispheric Ekman spiral, resulting in six combinations of rotor rotation and inflow wind condition. The wake strength, extension, width, and deflection depend on the interaction of the meridional component of Ekman spiral with the rotational direction of the actuator disc, whereas the direction of the disc rotation only marginally modifies the wake if no veer is present. The differences result from the amplification or weakening/reversion of the spanwise and the vertical wind components due to the effect of the superposed disc rotation. They are also present in the streamwise wind component of the wake and in the total turbulence intensity. In the case of an counterclockwise rotating actuator disc, the spanwise and vertical wind components increase directly behind the rotor, resulting in the same rotational direction in the whole wake while its strength decreases downwind. In the case of a clockwise rotating actuator disc, however, the spanwise and vertical wind components of the near wake are weakened or even reversed in comparison to the inflow. This weakening/reversion results in a downwind increase in the strength of the flow rotation in the wake or even a different rotational direction in the near wake in comparison to the far wake. The physical mechanism responsible for this difference can be explained by a simple linear superposition of a veering inflow with a Rankine vortex.