Organ morphogenesis is a complex process coordinated by cell specification, epithelial-mesenchymal interactions and tissue polarity. A striking example is the pattern of regularly spaced, globally aligned mammalian hair follicles, which emerges through epidermal-dermal signaling and planar polarized morphogenesis. Here, using live-imaging, we discover that developing hair follicles polarize through dramatic cell rearrangements organized in a counter-rotational pattern of cell flows. Upon hair placode induction, Shh signaling specifies a radial pattern of progenitor fates that, together with planar cell polarity, induce counter-rotational rearrangements through myosin and ROCK-dependent polarized neighbour exchanges. Importantly, these cell rearrangements also establish cell fate asymmetry by repositioning radial progenitors along the anterior-posterior axis. These movements concurrently displace associated mesenchymal cells, which then signal asymmetrically to maintain polarized cell fates. Our results demonstrate how spatial patterning and tissue polarity generate an unexpected collective cell behaviour that in turn, establishes both morphological and cell fate asymmetry.