Recent work has shown that the Milky Way and the Andromeda galaxies both possess the unexpected property that their dwarf satellite galaxies are aligned in thin and kinematically coherent planar structures. It is interesting to evaluate the incidence of such planar structures in the larger galactic population, because the Local Group may not be a representative environment. Here we report measurements of the velocities of pairs of diametrically opposed satellite galaxies. In the local Universe (redshift z < 0.05), we find that satellite pairs out to a distance of 150 kiloparsecs from the galactic centre are preferentially anti-correlated in their velocities (99.994 per cent confidence level), and that the distribution of galaxies in the larger-scale environment (out to distances of about 2 megaparsecs) is strongly clumped along the axis joining the inner satellite pair (>7σ confidence). This may indicate that planes of co-rotating satellites, similar to those seen around the Andromeda galaxy, are ubiquitous, and their coherent motion suggests that they represent a substantial repository of angular momentum on scales of about 100 kiloparsecs.