  The use of gyrokinetics, wherein phase-space coordinate transformations
result in a phase-space dimensionality reduction as well as removal of fast
time scales, has enabled the simulation of microturbulence in fusion devices.
The state-of-the-art gyrokinetic models used in practice are parallel-only
models wherein the perpendicular part of the vector potential is neglected.
Such models are inherently not gauge invariant. We generalize the work of
[Burby, Brizard. Physics Letters A, 383(18):2172-2175] by deriving a sufficient
condition on the gyrocentre coordinate transformation which ensures gauge
invariance. This leads to a parametrized family of gyrokinetic models for which
we motivate a specific choice of parameters that results in a physically
meaningful gauge invariant model. Due to gauge invariance this model can be
expressed directly in terms of the electromagnetic fields, rather than the
potentials, and the gyrokinetic model thereby results in the macroscopic
Maxwell's equations. For the linearized model, it is demonstrated that the
shear and compressional Alfv\'en waves are present with the correct
frequencies. The fast compressional Alfv\'en wave can be removed by making use
of a Darwin-like approximation. This approximation breaks gauge invariance, but
we find that the field equations are still compatible without the need of a
Lagrange multiplier.
