Galaxy Zoo is the first study of nearby galaxies that contains reliable information about the spiral sense of rotation of galaxy arms for a sizeable number of galaxies. We measure the correlation function of spin chirality (the sense in which galaxies appear to be spinning) of face-on spiral galaxies in angular, real and projected spaces. Our results indicate a hint of positive correlation at separations less than ∼0.5 Mpc at a statistical significance of 2σ–3σ. This is the first experimental evidence for chiral correlation of spins. Within the tidal torque theory, it indicates that the inertia tensors of nearby galaxies are correlated. This is complementary to the studies of nearby spin axis correlations that probe the correlations of the tidal field. Theoretical interpretation is made difficult by the small distances at which the correlations are detected, implying that substructure might play a significant role, and our necessary selection of face-on spiral galaxies, rather than a general volume-limited sample.