In this work, we introduce a new principle that provides a tool for searching for odd periodic responses of a general nonlinear oscillator with symmetries. This result arises as the dual version of the variational principle first introduced in Ortega (2016), because in this case the nonlinearity $ xD(t, x) $ verifies the reversed inequality, i.e., $ D(t, 0)<D(t, x) $ for all $ t\in {\mathbb{R}} $ and $ x\neq0 $. Thus, our result reveals that under certain conditions there exists a family of odd periodic responses with prescribed nodal properties for the general oscillator. Indeed, contrary to that obtained in Ortega (2016), the number of zeros of the solutions given by this dual principle is at least bounded below. To illustrate the application of our result, we consider a real example where the reversed inequality arises naturally: a noninterdigitated Comb-drive MEMS modeled by a nonlinear version of the Mathieu equation. Then, we prove the existence of a bi-stability operation regime for this example, since under certain conditions, the positive subharmonic of order 2 given by our principle and the trivial solution are linearly stable. Our results are based on classical ODE tools and the perturbation approach in Cen et al. (2020).