Nanoscale droplets on a substrate are of great interest because of their relevance for droplet-based technologies for light manipulation, lab-on-chip devices, miniaturized reactors, encapsulation, and many others. In this work, we establish a basic principle for symmetrical arrangements of surface nanodroplets during their growth out of oversaturated solution established through solvent exchange, which takes place under simple and controlled flow conditions. In our model system, nanodroplets nucleate at the rim of spherical cap microstructures on a substrate, due to a pulse of oversaturation supplied by a solvent exchange process. We find that, while growing at the rim of the microcap, the nanodroplets self-organize into highly symmetric arrangements, with respect to position, size, and mutual distance. The angle between the neighboring droplets is 4 times the ratio between the base radii of the droplets and the spherical caps. We show and explain how the nanodroplets acquire the symmetrical spatial arrangement during their competitive growth and why and how the competition enhances the overall growth rate of the nucleated nanodroplets. This mechanism behind the nanodroplet self-organization promises a simple approach to control the location of droplets with a volume down to attoliters.