We show that all purely magnetic Petrov type D, irrotational, aligned perfect fluid spacetimes are locally rotationally symmetric class III by the Ellis classification. This result is combined with a similar theorem for purely magnetic Petrov type D, shear-free, aligned perfect fluid spacetimes. For all purely magnetic locally rotationally symmetric class III perfect fluid spacetimes the pressure and energy density are both expressed in terms of a single function (of the timelike coordinate only). Therefore, these expressions define, implicitly, a nonlinear barotropic equation of state on intervals of this single function where the energy conditions are satisfied. However, a general expression is obtained for which the pressure is explicitly written as a function of the energy density. It is shown that in the asymptotic limit of large pressure and energy density the ratio between the pressure and energy density approaches the value 1/5. The only member of these spacetimes with a linear barotropic equation of state and a homothetic vector field is the purely magnetic Collins–Stewart spacetime.