Using photoemission electron microscopy, we study the magnetization orientation in single 5-25 nm iron particles coupled to a ferromagnetic cobalt support. We find a noncollinear alignment between the particle and substrate magnetization above a particle size of approximately 6 nm and a parallel alignment for smaller sizes. Numerical calculations reveal a transition from an exchange-dominated to an anisotropy-dominated regime on increasing the particle height: the smaller particles are in a single-domain collinear state while larger particles exhibit a spin-spiral magnetic structure determined by the magnetic anisotropy energy.