Magnetic resonance imaging (MRI) reconstruction has largely been dominated by deep neural networks (DNN); however, many state-of-the-art architectures use black-box structures, which hinder interpretability and improvement. Here, we propose an interpretable DNN architecture for self-supervised MRI reconstruction and denoising by directly parameterizing and learning the classical primal-dual splitting, dubbed LPDSNet. This splitting algorithm allows us to decouple the observation model from the signal prior. Experimentally, we show other interpretable architectures without this decoupling property exhibit failure in the self-supervised learning regime. We report state-of-the-art self-supervised joint MRI reconstruction and denoising performance and novel noise-level generalization capabilities, where in contrast black-box networks fail to generalize.