  We study `purely radiative' (div E = div H = 0) and geodesic perfect fluids
with non-constant pressure and show that the Bianchi class A perfect fluids can
be uniquely characterized --modulo the class of purely electric and
(pseudo-)spherically symmetric universes-- as those models for which the
magnetic and electric part of the Weyl tensor and the shear are simultaneously
diagonalizable. For the case of constant pressure the same conclusion holds
provided one also assumes that the fluid is irrotational.
