Intuition suggests that an object should carry all of its physical properties. However, a quantum object may not act in such a manner-it can temporarily leave some of its physical properties where it never appears. This phenomenon is known as the quantum Cheshire cat effect. It has been proposed that a quantum object can even permanently discard a physical property and obtain a new one it did not initially have. Here, we observe this effect experimentally by casting non-unitary imaginary-time evolution on a photonic cluster state to extract weak values, which reveals the counterintuitive phenomenon that two photons exchange their spins without classically meeting each other. A phenomenon presenting only in the quantum realm, our results are in stark contrast with the perception of inseparability between objects and properties, and shed new light on comprehension of the ontology of observables.