The goal of the paper is to study the asymptotic behavior of solutions to semilinear stochastic partial differential equations (PDEs) on a smooth bounded domain. More precisely, the authors prove the existence of invariant measures for the Markovian semigroup generated by the solution to a parabolic semilinear stochastic PDE whose nonlinear drift term satisfies only a kind of symmetry condition on its behavior at infinity, but no restriction on its growth rate is imposed. Thanks to strong integrability properties of invariant measures \(\mu\), the solvability of the associated Kolmogorov equation in the space \(L^1(\mu)\) is then established, and the infinitesimal generator of the transition semigroup is identified as the closure of the Kolmogorov operator. A key role is played by a generalized variational setting. If we compare with the existing literature, similar results were obtained under local Lipschitz-continuity or other suitable growth assumptions on the drift. Another positive aspect of the paper is that the authors use only standard monotonicity assumptions. Concerning the stochastic tools, a priori estimates (which, in turn, are obtained by stochastic calculus) and the generalized Itô formulas are used.