A methodology for deriving a pressure–strain correlation model with variable coefficients is developed. The methodology is based on two important premises: (i) the extreme states of turbulence – the rapid distortion and equilibrium limits – are more amenable to mathematically rigorous modelling because of significant simplifications not possible at other states; and (ii) the models of the extreme states collectively contain all of the relevant physics so that models for any intermediate state can be obtained by suitable interpolation. A pressure–strain model of the standard form is considered and the coefficients are determined from linear analysis in the rapid distortion limit and from a fixed point analysis in the equilibrium limit. The model coefficients, which depend on the mean deformation and turbulence state, vary from flow to flow in a manner consistent with Navier–Stokes physics.The exact causal relationship between the model coefficients and the model's equilibrium behaviour is established by fixed point analysis performed using representation theory. Then, the equilibrium values of the model coefficients are chosen to yield the observed equilibrium behaviour. The values of the model coefficients in the rapid distortion limit are determined by enforcing consistency with the Crow constraint. The new variable-coefficient model reduces to the traditional constant-coefficient model in strain-dominated turbulent flows near equilibrium. The model performance in bench-mark turbulent flows, in which the traditional models have been calibrated extensively, is preserved intact. The new model is significantly different from the traditional one in mean vorticity-dominated and non-equilibrium turbulence. These two important classes of flows, in which traditional models fail, are successfully captured by the new model.