We discuss two variants of a two-sided Jacobi–Davidson (JD) method, which have asymptotically cubic convergence for nonnormal matrices, and aim to find both right and left eigenvectors. These methods can be seen as Jacobi–Davidson analogs of Ostrowski's two-sided Rayleigh quotient iteration (RQI). Some relations between (exact and inexact) two-sided Jacobi–Davidson and (exact and inexact) two-sided Rayleigh quotient iteration are given, together with convergence rates. Furthermore, we introduce an alternating Jacobi–Davidson process that can be seen as the Jacobi–Davidson analog of Parlett's alternating Rayleigh quotient iteration. The methods are extended to the generalized and polynomial eigenproblem. Advantages of the methods are illustrated by numerical examples.