A theory of self-gravitating, pressure truncated, rotating molecular filaments with helical magnetic fields is developed to model molecular tornadoes recently observed in the Galactic Centre. A Monte Carlo exploration of the model parameter space is constrained by observations of three molecular tornadoes via three parameters involving observable quantities. It is found that torsional Alfvén wave physics is an important condition in realistic models. A virial analysis indicates that the magnetic field component dominates the pressure component, and both dominate self-gravity. “Phase diagrams” of numerical solutions to rotating, self-gravitating, unmagnetized cylinders are constructed to better understand solutions with “density inversions” found in the molecular tornado models. By varying free parameters of rotation and equation of state (EOS), five solution types are found and categorized. Self-truncating solutions with density inversions are found to occupy a larger portion of the solution space around the isothermal EOS.