We perform a linear stability analysis of magnetized rotating cylindrical jet flows in the approximation of zero thermal pressure. We focus our analysis on the effect of rotation on the current driven mode and on the unstable modes introduced by rotation. We find that rotation has a stabilizing effect on the current driven mode only for rotation velocities of the order of the Alfvén velocity. Rotation introduces also a new unstable centrifugal buoyancy mode and the ‘cold’ magnetorotational instability. The first mode is analogous to the Parker instability with the centrifugal force playing the role of effective gravity. The magnetorotational instability can be present, but only in a very limited region of the parameter space and is never dominant. The current driven mode is characterized by large wavelengths and is dominant at small values of the rotational velocity, while the buoyancy mode becomes dominant as rotation is increased and is characterized by small wavelengths.