Significance Living systems differ from dead matter in one crucial aspect: They are driven by internal engines. In anisotropic fluids, the widely accepted framework of active matter theory predicts that this necessarily leads to an instability from quiescence to large-scale and eventually incoherent motion. Here we challenge this common wisdom by reexamining the symmetries of the most ubiquitous experimental geometry for active systems. We uncover an additional coupling between activity and motion that can make an active system even more stable than its passive counterpart, while preserving other hallmarks of nonequilibrium physics. Our results challenge common views on active matter and clarify the interpretation of multiple experiments.