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It was shown recently that entanglement of identical particles has a feature called dualism (Bose and Home 2013 Phys. Rev. Lett. http://dx.doi.org/10.1103/PhysRevLett.110.265701 110 http://dx.doi.org/10.1103/PhysRevLett.110.265701 ), which is fundamentally connected with quantum indistinguishability. Here we report an experiment that observes the entanglement duality for the first time with two identical photons, which manifest polarization entanglement when labeled by different paths or path entanglement when labeled by polarization states. By adjusting the mismatch in frequency or arrival time of the entangled photons, we tune the photon indistinguishability from the quantum to the classical limit and observe that the entanglement duality disappears under the emergence of classical distinguishability, confirming it as a characteristic feature of quantum indistinguishable particles.