We consider the characterization of global attractors $A_f$ for semiflows generated by scalar one-dimensional semilinear parabolic equations of the form $u_t = u_{xx} + f(u,u_x)$, defined on the circle $x\in S^1$, for a class of reversible nonlinearities. We modify a proof developed for nonlinearities of simple type, making it simpler and amenable to generalization. We obtain a classification up to connection equivalence of global attractors for $S^1$-equivariant parabolic equations.