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Clouds play an important role in the Earth's energy budget and their behavior is one of the largest uncertainties in future climate projections. Satellite observations should help in understanding cloud responses, but decades and petabytes of multispectral cloud imagery have to date received only limited use. This study reduces the dimensionality of satellite cloud observations by grouping them via a novel automated, unsupervised cloud classification technique by using a convolutional neural network. Our technique combines a rotation-invariant autoencoder with hierarchical agglomerative clustering to generate cloud clusters that capture meaningful distinctions among cloud textures, using only raw multispectral imagery as input. Thus, cloud classes are defined without reliance on location, time/season, derived physical properties, or pre-designated class definitions. We use this approach to generate a unique new cloud dataset, the AI-driven cloud classification atlas (AICCA), which clusters 22 years of ocean images from the Moderate Resolution Imaging Spectroradiometer (MODIS) on NASA's Aqua and Terra instruments - 800 TB of data or 198 million patches roughly 100 km x 100 km (128 x 128 pixels) - into 42 AI-generated cloud classes. We show that AICCA classes involve meaningful distinctions that employ spatial information and result in distinct geographic distributions, capturing, for example, stratocumulus decks along the West coasts of North and South America. AICCA delivers the information in multi-spectral images in a compact form, enables data-driven diagnosis of patterns of cloud organization, provides insight into cloud evolution on timescales of hours to decades, and helps democratize climate research by facilitating access to core data.