One of the greatest sources of uncertainty in future climate projections comes from limitations in modelling clouds and in understanding how different cloud types interact with the climate system. A key first step in reducing this uncertainty is to accurately classify cloud types at high spatial and temporal resolution. In this paper, we introduce Cumulo, a benchmark dataset for training and evaluating global cloud classification models. It consists of one year of 1km resolution MODIS hyperspectral imagery merged with pixel-width 'tracks' of CloudSat cloud labels. Bringing these complementary datasets together is a crucial first step, enabling the Machine-Learning community to develop innovative new techniques which could greatly benefit the Climate community. To showcase Cumulo, we provide baseline performance analysis using an invertible flow generative model (IResNet), which further allows us to discover new sub-classes for a given cloud class by exploring the latent space. To compare methods, we introduce a set of evaluation criteria, to identify models that are not only accurate, but also physically-realistic.