In turbulent rotating convection a typical flow-structuring in columnar vortices is observed. In the internal structure of these vortices several symmetries are approximately satisfied. A model for these columnar vortices is derived by prescribing these symmetries. The symmetry constraints are applied to the Navier-Stokes equations with rotation in the Boussinesq approximation. It is found that the application of the symmetries results in a set of linearized equations. An investigation of the linearized equations leads to a model for the columnar vortices, and a prediction for the heat flux (Nusselt number) that is very appropriate compared with results from direct numerical simulations of the full governing equations.