This paper presents a new \emph{bi-Lipschitz} invertible neural network, the BiLipNet, which has the ability to control both its \emph{Lipschitzness} (output sensitivity to input perturbations) and \emph{inverse Lipschitzness} (input distinguishability from different outputs). The main contribution is a novel invertible residual layer with certified strong monotonicity and Lipschitzness, which we compose with orthogonal layers to build bi-Lipschitz networks. The certification is based on incremental quadratic constraints, which achieves much tighter bounds compared to spectral normalization. Moreover, we formulate the model inverse calculation as a three-operator splitting problem, for which fast algorithms are known. Based on the proposed bi-Lipschitz network, we introduce a new scalar-output network, the PLNet, hich satisfies the Polyak-\L{}ojasiewicz condition. It can be applied to learn non-convex surrogate losses with favourable properties, e.g., a unique and efficiently-computable global minimum.