  We report the experimental violation of Leggett-type inequalities for a
hybrid entangled state of spin and orbital angular momentum of a single photon.
These inequalities give a physical criterion to verify the possible validity of
a class of hidden-variable theories, originally named "crypto non-local", that
are not excluded by the violation of Bell-type inequalities. In our case, the
tested theories assume the existence of hidden variables associated with
independent degrees of freedom of the same particle, while admitting the
possibility of an influence between the two measurements, i.e. the so-called
contextuality of observables. We observe a violation the Leggett inequalities
for a range of experimental inputs, with a maximum violation of seven standard
deviations, thus ruling out this class of hidden variable models with a high
confidence.
