Collective cell migration plays an essential role in various biological processes, such as development or cancer proliferation. While cell-cell interactions are clearly key determinants of collective cell migration, the physical mechanisms that control the emergence of cell clustering and collective cell migration are still poorly understood. In particular, observations have shown that binary cell-cell collisions generally lead to antialignment of cell polarities and separation of pairs-a process called contact inhibition of locomotion (CIL), which is expected to disfavor the formation of large-scale cell clusters with coherent motion even though the latter is often observed in tissues. To solve this puzzle, we adopt a joint experimental and theoretical approach to determine the large-scale dynamics of cell assemblies from elementary pairwise cell-cell interaction rules. We quantify experimentally binary cell-cell interactions and show that they can be captured by a minimal equilibriumlike pairwise asymmetric aligning interaction potential that reproduces the CIL phenomenology. We identify its symmetry class, build the corresponding active hydrodynamic theory, and show on general grounds that such asymmetric aligning interaction destroys large-scale clustering and ordering, leading instead to a liquidlike microphase of cell clusters of finite size and short lived polarity or to a fully dispersed isotropic phase. Finally, this shows that CIL-like asymmetric interactions in cellular systems-or general active systems-control cluster sizes and polarity, and can prevent large-scale coarsening and long-range polarity, except in the singular regime of dense confluent systems.