We show that there are no new consistent cosmological perfect fluid solutions when in an open neighbourhood U of an event the fluid kinematical variables and the electric and magnetic Weyl curvature are all assumed to be rotationally symmetric about a common spatial axis, specializing the Weyl curvature tensor to algebraic Petrov type D. The consistent solutions of this kind are either locally rotationally symmetric, or are subcases of the Szekeres dust models. Parts of our results require the assumption of a barotropic equation of state. Additionally we demonstrate that local rotational symmetry of perfect fluid cosmologies follows from rotational symmetry of the Riemann curvature tensor and of its covariant derivatives only up to second order, thus strengthening a previous result.