This paper is related to the study of systems of reflected backward stochastic differential equations with interconnected bilateral obstacles. These systems are connected with zero-sum stochastic switching games. Under appropriate assumptions, we provide either existence or existence and uniqueness of the solution of those systems when the switching costs are Itˆo processes. The link with systems of PDEs with bilateral interconnected obstacles is also stated via the Feynman-Kac representation when randomness comes from a Markov diffusion process.