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The quantum anomalous Hall (QAH) effect has been recently discovered in an experiment using a thin-film topological insulator with ferromagnetic ordering and strong spin-orbit coupling. Here we investigate the spin degree of freedom of a QAH insulator and uncover the fundamental phenomenon that the edge states exhibit a topologically stable spin texture in the boundary when a chiral-like symmetry is present. This result shows that edge states are chiral in both the orbital and spin degrees of freedom, and the chiral edge spin texture corresponds to the bulk topological states of the QAH insulator. We also study the potential applications of the edge spin texture in designing topological-state-based spin devices, which might be applicable to future spintronic technologies.