Predictive clustering trees (PCTs) are a well established generalization of standard decision trees, which can be used to solve a variety of predictive modeling tasks, including structured output prediction. Combining them into ensembles yields state-of-the-art performance. Furthermore, the ensembles of PCTs can be interpreted by calculating feature importance scores from the learned models. However, their learning time scales poorly with the dimensionality of the output space. This is often problematic, especially in (hierarchical) multi-label classification, where the output can consist of hundreds of potential labels. Also, learning of PCTs can not exploit the sparsity of data to improve the computational efficiency, which is common in both input (molecular fingerprints, bag of words representations) and output spaces (in multi-label classification, examples are often labeled with only a fraction of possible labels). In this paper, we propose oblique predictive clustering trees, capable of addressing these limitations. We design and implement two methods for learning oblique splits that contain linear combinations of features in the tests, hence a split corresponds to an arbitrary hyperplane in the input space. The methods are efficient for high dimensional data and capable of exploiting sparse data. We experimentally evaluate the proposed methods on 60 benchmark datasets for 6 predictive modeling tasks. The results of the experiments show that oblique predictive clustering trees achieve performance on-par with state-of-the-art methods and are orders of magnitude faster than standard PCTs. We also show that meaningful feature importance scores can be extracted from the models learned with the proposed methods.