We present a family of self-consistent axisymmetric rotating globular cluster models which are fitted to spectroscopic data for NGC 362, NGC 1851, NGC 2808, NGC 4372, NGC 5927 and NGC 6752 to provide constraints on their physical and kinematic properties, including their rotation signals. They are constructed by flattening Modified Plummer profiles, which have the same asymptotic behaviour as classical Plummer models, but can provide better fits to young clusters due to a slower turnover. The models are in dynamical equilibrium as they depend solely on the action variables. We employ a fully Bayesian scheme to investigate the uncertainty in our model parameters (including mass-to-light ratios and inclination angles) and evaluate the Bayesian evidence ratio for rotating to non-rotating models. We find convincing levels of rotation only in NGC 2808. In the other clusters, there is only a hint of rotation (in particular, NGC 4372 and NGC 5927), as the data quality does not allow us to draw strong conclusions. Where rotation is present, we find that it is confined to the central regions. As part of this work, we have developed a novel q-Gaussian basis expansion of the line-of-sight velocity distributions, from which general models can be constructed via interpolation on the basis coefficients.