We present NeuralBranch, an interpretable neural network framework. In this work, we use it specifically to predict the pedestal from key engineering parameters in tokamak fusion experiments. The main goal is to uncover intricate relationships that traditional power scalings, with their limited expressive capacity, fail to capture. A secondary objective is to provide a transparent alternative to current opaque, black-box machine learning models used to predict the pedestal in integrated modeling frameworks. By using the proposed method, we obtain a novel global overview of several intricate dependencies in the JET pedestal database. For instance, while both input power and plasma current are positively correlated with pedestal top pressure and temperature, NeuralBranch reveals an attenuating interaction. This means that increasing power weakens the impact that current has on pedestal pressure and temperature, and vice versa. Further investigation of this interaction may be important to avoid overestimating pedestal stored energy at future machines like ITER when using established power scalings. We also identify an amplifying interaction between plasma current and triangularity, where higher triangularity amplifies the effect of plasma current on pedestal density, and vice versa. In addition to these findings, NeuralBranch matches the accuracy of black-box neural networks, with R2 values as high as 0.88. This demonstrates that interpretability, with its associated benefits, can be achieved without sacrificing accuracy, making NeuralBranch a promising alternative for pedestal predictions.