Countless processes in nature and industry, from rain droplet nucleation to plankton interaction in the ocean, are intimately related to turbulent fluctuations of local concentrations of advected matter. These fluctuations can be described by considering the change of the separation between particle pairs, known as pair dispersion, which is believed to obey a cubic in time growth according to Richardson's theory. Our work reveals a universal, scale-invariant alignment between the relative velocity and position vectors of dispersing particles at a mean angle that we show to be a universal constant of turbulence. We connect the value of this mean angle to Richardson's traditional theory and find agreement with data from a numerical simulation and a laboratory experiment. While the Richardson's cubic regime has been observed for small initial particle separations only, the constancy of the mean angle manifests throughout the entire inertial range of turbulence. Thus, our work reveals the universal nature of turbulent pair dispersion through a geometrical paradigm whose validity goes beyond the classical theory, and provides a framework for understanding and modeling transport and mixing processes.