We consider a real planar analytic vector field, X, such that the origin, O, is a centre for the linearization of X. Poincare's condition of reversibility with respect to a line passing through O is then a sufficient condition for O to be a centre for the vector field X. We provide necessary and sufficient conditions, involving the vanishing of certain polynomials in the coefficients in the expansion of X, for Ž . reversibility. We also show that if the linearization, L x , of the divergence of X is Ž . non-trivial, then the only possible reversibility line is given by L x s 0; in such cases, this provides the basis for a simple test of reversibility. We examine the consequences of our various tests for quadratic and cubic vector fields; all non-Ž . Hamiltonian cases are discussed. When L x ' 0 in cubic systems, it is possible for Ž . the reversibility line if it exists to be unique, but it is also possible for there to be two such lines. These possibilities are characterized algebraically, and a prescrip-Ž . tion is provided for determining the reversibility line s in each case.