  We consider the dynamics of a vibrating and rotating end-mirror of an optical
Fabry-P{\'erot} cavity that can sustain Laguerre-Gaussian modes. We demonstrate
theoretically that since the intra-cavity field carries linear as well as
angular momentum, radiation pressure can create bipartite entanglement between
a vibrational and a rotational mode of the mirror. Further we show that the
ratio of vibrational and rotational couplings with the radiation field can
easily be adjusted experimentally, which makes the generation and detection of
entanglement robust to uncertainties in the cavity manufacture. This
constitutes the first proposal to demonstrate entanglement between two
qualitatively different degrees of freedom of the same macroscopic object.
