  It is shown that the zilch conservation law arises as the Noether current
corresponding to a variational symmetry of a duality-symmetric Maxwell
Lagrangian. The action of the corresponding symmetry generator on the
duality-symmetric Lagrangian, while non-vanishing, is a total divergence as
required by the Noether theory. The variational nature of the zilch
conservation law was previously known only for some of the components of the
zilch tensor, notably the optical chirality. By contrast, our analysis is fully
covariant and is therefore valid for all components of the zilch tensor. The
analysis is presented here for both real and complex versions of
duality-symmetric Maxwell Lagrangians.
