Many epithelia have a common planar cell polarity (PCP), as exemplified by the consistent orientation of hairs on mammalian skin and insect cuticle. One conserved system of PCP depends on Starry night (Stan, also called Flamingo), an atypical cadherin that forms homodimeric bridges between adjacent cells. Stan acts together with other transmembrane proteins, most notably Frizzled (Fz) and Van Gogh (Vang, also called Strabismus). Here, using an in vivo assay for function, we show that the quintessential core of the Stan system is an asymmetric intercellular bridge between Stan in one cell and Stan acting together with Fz in its neighbour: such bridges are necessary and sufficient to polarise hairs in both cells, even in the absence of Vang. By contrast, Vang cannot polarise cells in the absence of Fz; instead, it appears to help Stan in each cell form effective bridges with Stan plus Fz in its neighbours. Finally, we show that cells containing Stan but lacking both Fz and Vang can be polarised to make hairs that point away from abutting cells that express Fz. We deduce that each cell has a mechanism to estimate and compare the numbers of asymmetric bridges, made between Stan and Stan plus Fz, that link it with its neighbouring cells. We propose that cells normally use this mechanism to read the local slope of tissue-wide gradients of Fz activity, so that all cells come to point in the same direction.