We classify the rational differential 1-forms with simple poles and simple zeros on the Riemann sphere according to their isotropy group; when the 1-form has exactly two poles the isotropy group is isomorphic to C * , namely {z → az | a ∈ C, a = 0}, and when the 1-form has k ≥ 3 poles the isotropy group is finite. In particular we show that all the finite subgroups of P SL(2, C) are realizable as isotropy groups for a rational 1-form on C. We also present local and global geometrical conditions for their classification. The classification result enables us to describe the moduli space of rational 1-forms with finite isotropy that have exactly k simple poles and k − 2 simple zeros on the Riemann sphere. Moreover, we provide sufficient (geometrical) conditions for when the 1-forms are isochronous. Concerning the recent work of J.C. Langer, we reflect on the strong relationship between our work and his and provide a partial answer regarding polyhedral geometries that arise from rational quadratic differentials on the Riemann sphere.