In this paper, we study systems of nonlinear second-order variational inequalities with interconnected bilateral obstacles with non-local terms. They are of min–max and max–min types and related to a multiple modes zero-sum switching game in the jump-diffusion model. Using systems of penalized reflected backward SDEs with jumps and unilateral interconnected obstacles, and their associated deterministic functions, we construct for each system a continuous viscosity solution which is unique in the class of functions with polynomial growth.