  In guiding center theory, the standard gyro-angle coordinate is associated
with gyro-gauge dependence, the global existence problem for unit vectors
perpendicular to the magnetic field, and the notion of anholonomy, which is the
failure of the gyro-angle to return to its original value after being
transported around a loop in configuration space. We analyse these three
intriguing topics through the lens of a recently proposed, global,
gauge-independent gyro-angle. This coordinate is constrained, and therefore
necessitates the use of a covariant derivative. It also highlights the
intrinsic meaning and physical content of gyro-gauge freedom and anholonomy.
There are, in fact, many possible covariant derivatives compatible with the
intrinsic gyro-angle, and each possibility corresponds to a different notion of
gyro-angle transport. This observation sheds new light on Littlejohn's notion
of gyro-angle transport and suggests a new derivation of the
recently-discovered global existence condition for unit vectors perpendicular
to the magnetic field. We also discuss the relationship between Cartesian
position-momentum coordinates and the intrinsic gyro-angle.
