We consider oscillatory flows between concentric co-rotating cylinders at angular velocity Ω(t) = Ωm + Ωo cos ωt as a prototype to investigate the competing effects of centrifugal and Coriolis forces on the flow stability. We first study by flow visualization the effect of the mean rotation Ωm on the centrifugal destabilization due to the temporal modulation. We show that increasing the mean rotation first destabilizes and then restabilizes the flow. The instability of the purely azimuthal basic flow is then analysed by investigating the dynamics of the axial velocity component of the vortex structures. Velocity measurements performed in the rotating frame of the cylinders using ultrasound Doppler velocimetry show that secondary flow appears and disappears several times during a flow period. Based on a finite-gap expression for the basic flow, linear stability analysis is performed with a quasi-steady approach, providing the times of appearance and disappearance of secondary flow in a cycle as well as the effect on the instability threshold of the mean rotation. The theoretical and numerical results are in agreement with experimental results up to intermediate values of the frequency. Notably, the flow periodically undergoes restabilization at particular time intervals.