We present an experimental study of ‘polygons’ forming on the free surface of a swirling water flow in a partially filled cylindrical container. In our set-up, we rotate the bottom plate and the cylinder wall with separate motors. We thereby vary rotation rate and shear strength independently and move from a rigidly rotating ‘Newton’s bucket’ flow to one where bottom and cylinder wall are rotating oppositely and the surface is strongly turbulent but flat on average. Between those two extremes, we find polygonal states for which the rotational symmetry is spontaneously broken. We investigate the phase diagram spanned by the two rotational frequencies at a given water filling height and find polygons in a regime, where the two frequencies are sufficiently different and, predominantly, when they have opposite signs. In addition to the extension of the family of polygons found with the stationary cylinder, we find a new family of smaller polygons for larger rotation rates of the cylinder, opposite to that of the bottom plate. Further, we find a ‘monogon’, a figure with one corner, roughly an eccentric circle rotating in the same sense as the cylinder. The case where only the bottom plate is rotating is compared with the results of Jansson et al. (Phys. Rev. Lett., vol. 96, 2006, art. 174502), where the same size of cylinder was used, and although the overall structure of the phase diagram spanned by water height and rotational frequency is the same, many details are different. To test the effect of small experimental defects, such as misalignment of the bottom plate, we investigate whether the rotating polygons are phase locked with the bottom plate, and although we find cases where the frequency ratio of figure and bottom plate is nearly rational, we do not find phase locking. Finally, we show that the system has a surprising multistability and excitability, and we note that this can cause quantitative differences between the phase diagrams obtained in comparable experiments.