  A fully relativistically covariant and manifestly gauge invariant formulation
of classical Maxwell electrodynamics is presented, purely in terms of gauge
invariant potentials without entailing any gauge fixing. We show that the
inhomogeneous equations satisfied by the physical scalar and vector potentials
(originally discovered by Maxwell) have the same symmetry as the isometry of
Minkowski spacetime, thereby reproducing Einstein's incipient approach leading
to his discovery of special relativity as a spacetime symmetry. To arrive at
this conclusion, we show how the Maxwell equations for the potentials follow
from stationary electromagnetism by replacing the Laplacian operator by the
d'Alembertian operator, while making all variables dependent on space and time.
We also establish consistency of these equations by deriving them from the
standard Maxwell equations for the field strengths, showing that there is a
unique projection operator which projects onto the physical potentials.
Properties of the physical potentials are elaborated through their iterative
N\"other coupling to a charged scalar field leading to the Abelian Higgs model,
and through a sketch of the Aharonov-Bohm effect, where dependence of the
Aharonov-Bohm phase on the physical vector potential is highlighted.
