The ring-like cohesin complex mediates sister chromatid cohesion by encircling pairs of sister chromatids. Cohesin also extrudes loops along chromatids. Whether the two activities involve similar mechanisms of DNA engagement is not known. We implemented an experimental approach based on isolated nuclei carrying engineered cleavable RAD21 proteins to precisely control cohesin ring integrity so that its role in chromatin looping could be studied under defined experimental conditions. This approach allowed us to identify cohesin complexes with distinct biochemical, and possibly structural properties, that mediate different sets of chromatin loops. When RAD21 is cleaved and the cohesin ring is opened, cohesin complexes at CTCF sites are released from DNA and loops at these elements are lost. In contrast, cohesin-dependent loops within chromatin domains and that are not anchored at CTCF sites are more resistant to RAD21 cleavage. The results show that the cohesin complex mediates loops in different ways depending on genomic context and suggests that it undergoes structural changes as it dynamically extrudes and encounters CTCF sites.