Bilateral symmetric tissues must interpret axial references to maintain their global architecture during growth or repair. The regeneration of hair cells in the zebrafish lateral line, for example, forms a vertical midline that bisects the neuromast epithelium into perfect mirror-symmetric plane-polarized halves. Each half contains hair cells of identical planar orientation but opposite to that of the confronting half. The establishment of bilateral symmetry in this organ is poorly understood. Here, we show that hair-cell regeneration is strongly directional along an axis perpendicular to that of epithelial planar polarity. We demonstrate compartmentalized Notch signaling in neuromasts, and show that directional regeneration depends on the development of hair-cell progenitors in polar compartments that have low Notch activity. High-resolution live cell tracking reveals a novel process of planar cell inversions whereby sibling hair cells invert positions immediately after progenitor cytokinesis, demonstrating that oriented progenitor divisions are dispensable for bilateral symmetry. Notwithstanding the invariably directional regeneration, the planar polarization of the epithelium eventually propagates symmetrically because mature hair cells move away from the midline towards the periphery of the neuromast. We conclude that a strongly anisotropic regeneration process that relies on the dynamic stabilization of progenitor identity in permissive polar compartments sustains bilateral symmetry in the lateral line.