  In this paper, we discuss Galilean relativistic Maxwell theory in detail. We
first provide a set of mapping relations, derived systematically, that connect
the covariant and contravariant vectors in the Lorentz relativistic and
Galilean relativistic formulations. Exploiting this map, we construct the two
limits of Galilean relativistic Maxwell theory from usual Maxwell's theory in
the potential formalism for both contravariant and covariant vectors which are
now distinct entities. Field equations are derived and their internal
consistency is shown. The entire analysis is then performed in terms of
electric and magnetic fields for both covariant and contravariant components.
Duality transformations and their connection with boost symmetry are discussed
which reveal a rich structure. The notion of twisted duality is introduced.
Next we consider gauge symmetry, construct Noether currents and show their
on-shell conservation. We also discuss shift symmetry under which the
Lagrangian is invariant, where the corresponding currents are now on-shell
conserved. At the end we analyse the theory by including sources for both
contravariant and covariant sectors. We show that sources are now off-shell
conserved
