Symbolic Regression (SR) searches for a closed-form mathematical expression describing the relationship between input and output features in data. The main theoretical draw of SR compared to traditional black-box regression techniques is that the learned models should be interpretable by design. However, typical SR methods struggle to discover sparse and accurate models when the shape of the output varies locally, depending on the values of some input features. Given that this is a common occurence in physics, SR should be able to learn piecewise models. We introduce a new piecewise SR framework called Unified Piecewise Symbolic Regression (UPSR). UPSR simultaneously partitions the input space into subregions and learns local regressors for each subregion, forming a global model unifying all subregions. We demonstrate its effectiveness on a large synthetic SR benchmark containing both piecewise and non-piecewise data structures. UPSR is shown to outperform state-of-the-art piecewise SR approaches, both qualitatively and quantitatively.