The angle between subsequent particle displacement increments is evaluated as a function of the time lag in isotropic turbulence. It is shown that the evolution of this angle contains two well-defined power laws, reflecting the multiscale dynamics of high-Reynolds number turbulence. The probability density function of the directional change is shown to be self-similar and well approximated by an analytically derived model assuming Gaussianity and independence of the velocity and the Lagrangian acceleration.