Modern data-driven tools are transforming application-specific polymer development cycles. Surrogate models that can be trained to predict properties of polymers are becoming commonplace. Nevertheless, these models do not utilize the full breadth of the knowledge available in datasets, which are oftentimes sparse; inherent correlations between different property datasets are disregarded. Here, we demonstrate the potency of multi-task learning approaches that exploit such inherent correlations effectively. Data pertaining to 36 different properties of over 13,000 polymers are supplied to deep-learning multi-task architectures. Compared to conventional single-task learning models, the multi-task approach is accurate, efficient, scalable, and amenable to transfer learning as more data on the same or different properties become available. Moreover, these models are interpretable. Chemical rules, that explain how certain features control trends in property values, emerge from the present work, paving the way for the rational design of application specific polymers meeting desired property or performance objectives.