High-resolution imaging has revolutionized the study of single cells in their spatial context. However, summarizing the great diversity of complex cell shapes found in tissues and inferring associations with other single-cell data remains a challenge. Here, we present CAJAL, a general computational framework for the analysis and integration of single-cell morphological data. By building upon metric geometry, CAJAL infers cell morphology latent spaces where distances between points indicate the amount of physical deformation required to change the morphology of one cell into that of another. We show that cell morphology spaces facilitate the integration of single-cell morphological data across technologies and the inference of relations with other data, such as single-cell transcriptomic data. We demonstrate the utility of CAJAL with several morphological datasets of neurons and glia and identify genes associated with neuronal plasticity inC. elegans. Our approach provides an effective strategy for integrating cell morphology data into single-cell omics analyses.