  In the guiding center theory, smooth unit vectors perpendicular to the
magnetic field are required to define the gyrophase. The question of global
existence of these vectors is addressed using a general result from the theory
of characteristic classes. It is found that there is, in certain cases, an
obstruction to global existence. In these cases, the gyrophase cannot be
defined globally. The implications of this fact on the basic structure of the
guiding center theory are discussed. In particular it is demonstrated that the
guiding center asymptotic expansion of the equations of motion can still be
performed in a globally consistent manner when a single global convention for
measuring gyrophase is unavailable. The latter fact is demonstrated directly by
deriving a new expression for the guiding-center Poincar\'e-Cartan form
exhibiting no dependence on the choice of perpendicular unit vectors.
