Sister chromatid cohesion requires cohesin to act as a protein linker to hold chromatids together. How cohesin tethers chromatids remains poorly understood. We have used optical tweezers to visualize cohesin as it holds DNA molecules. We show that cohesin complexes tether DNAs in the presence of Scc2/Scc4 and ATP demonstrating a conserved activity from yeast to humans. Cohesin forms two classes of tethers: a “permanent bridge” resisting forces over 80 pN and a force-sensitive “reversible bridge.” The establishment of bridges requires physical proximity of dsDNA segments and occurs in a single step. “Permanent” cohesin bridges slide when they occur in trans, but cannot be removed when in cis. Therefore, DNAs occupy separate physical compartments in cohesin molecules. We finally demonstrate that cohesin tetramers can compact linear DNA molecules stretched by very low force (below 1 pN), consistent with the possibility that, like condensin, cohesin is also capable of loop extrusion.