In this paper we consider the zilch conservation laws for Maxwell theory and demonstrate that in the duality-symmetric version of Maxwell theory, the zilch arises as a Noether current for a variational symmetry of the duality symmetric Lagrangian which we identify through an application of the reverse of the Noether theorem. A variational symmetry leaves Lagrangian invariant up to a total divergence, without restricting to solutions of the field equations. This fact was previously known only for the so-called chirality current, i.e. the 00-component of zilch.