We study the possible rotation of cluster galaxies, developing, testing, and applying a novel algorithm which identifies rotation, if such does exist, as well as its rotational centre, its axis orientation, rotational velocity amplitude, and, finally, the clockwise or counterclockwise direction of rotation on the plane of the sky. To validate our algorithms we construct realistic Monte Carlo mock rotating clusters and confirm that our method provides robust indications of rotation. We then apply our methodology on a sample of Abell clusters with z<=0.1 with member galaxies selected from the Sloan Digital Sky Survey DR10 spectroscopic data base. After excluding a number of substructured clusters, which could provide erroneous indications of rotation, and taking into account the expected fraction of misidentified coherent substructure velocities for rotation, provided by our Monte Carlo simulation analysis, we find that ~23 per cent of our clusters are rotating under a set of strict criteria. Loosening the strictness of the criteria, on the expense of introducing spurious rotation indications, we find this fraction increasing to ~28 per cent. We correlate our rotation indicators with the cluster dynamical state, provided either by their Bautz-Morgan type or by their X-ray isophotal shape and find for those clusters showing rotation within 1.5h^-1^_70_ Mpc that the significance of their rotation is related to the dynamically younger phases of cluster formation but after the initial anisotropic accretion and merging has been completed. Finally, finding rotational modes in galaxy clusters could lead to the necessity of correcting the dynamical cluster mass calculations.