  In 1964, Lipkin discovered the zilches, a set of conserved quantities in free
electromagnetism. Among the zilches, optical chirality was identified by Tang
and Cohen in 2010, serving as a measure of the handedness of light and leading
to investigations into light's interactions with chiral matter. While the
symmetries underlying the conservation of the zilches have been examined, the
derivation of zilch conservation laws from symmetries of the standard free
electromagnetic (EM) action using Noether's theorem has only been addressed in
the case of optical chirality. We provide the full answer by demonstrating that
the zilch symmetry transformations of the four-potential, $A_{\mu}$, preserve
the standard free EM action. We also show that the zilch symmetries belong to
the enveloping algebra of a "hidden" invariance algebra of free Maxwell's
equations. This "hidden" algebra is generated by familiar conformal
transformations and certain "hidden" symmetry transformations of $A_{\mu}$.
Generalizations of the ``hidden'' symmetries are discussed in the presence of a
material four-current, as well as in the theory of a complex Abelian gauge
field. Additionally, we extend the zilch symmetries of the standard free EM
action to the standard interacting action (with a non-dynamical four-current),
allowing for a new derivation of the continuity equation for optical chirality
in the presence of electric charges and currents. Furthermore, new continuity
equations for the remaining zilches are derived.
