A recent neutron interferometry experiment claims to demonstrate a paradoxical phenomenon dubbed the "quantum Cheshire cat" [Nat. Commun.5, 4492 (2014)]. We have reproduced and extended these results with an equivalent optical interferometer. The results suggest that the photon travels through one arm of the interferometer, while its polarization travels through the other. However, we show that these experimental results belong to the domain where quantum and classical wave theories coincide; there is nothing uniquely quantum about the illusion of this Cheshire cat.