The development of a three-dimensional viscous incompressible flow generated behind an infinitely long circular cylinder, impulsively started into rectilinear motion and rotationally oscillating, is studied computationally. The numerical scheme, an hybrid vortex method, is used to integrate the velocity–vorticity formulation of the Navier–Stokes equations. The Reynolds number considered is $\Rey\,{=}\,400$, which is moderate though beyond the critical values $\Rey_2\,{\simeq}\,190$ and $\Rey_2'\,{\simeq}\,260$ for which the flow becomes spontaneously three-dimensional. The numerical method is explained and its main points are developed. This scheme is then applied to solve some two-dimensional problems, both in order to validate the method and to compute a nominal two-dimensional flow, required to measure the impact of three-dimensionality. The three-dimensional flow past a steady cylinder is also compared to benchmark simulations. Once the flow has become fully three-dimensional, beyond the transient regime and saturation of instabilities, the cylinder begins a rotary oscillation around its axis. Two kinds of rotations are considered: constant amplitude and several frequencies, and constant frequency and various amplitudes. When amplitude and frequency are high enough, the whole flow comes back to its two-dimensional state. This result gives a justification for two-dimensional computations in the literature related to rotating cylinders. For the first super-harmonic frequency of the flow, a parametric study is performed in order to find the impact of the amplitude on the topology of the flow. A bifurcation is clearly identified. Finally, the mechanisms involved in the return to a two-dimensional state are explained: the interaction between transverse instabilities and von Kármán streets is quantified by means of a correlation analysis.