  We elaborate on the duality-symmetric nonlinear electrodynamics in a new
formulation with auxiliary tensor fields. The Maxwell field strength appears
only in bilinear terms of the corresponding generic Lagrangian, while the
self-interaction is presented by a function E depending on the auxiliary
fields. Two types of dualities inherent in the nonlinear electrodynamics models
admit a simple off-shell characterization in terms of this function. In the
standard formulation, the continuous U(1) duality symmetry is nonlinearly
realized on the Maxwell field strength. In the new setting, the same symmetry
acts as linear U(1) transformations of the auxiliary field variables. The
nonlinear U(1) duality condition proves to be equivalent to the linear U(1)
invariance of the self-interaction E. The discrete self-duality (or
self-duality by Legendre transformation) amounts to a weaker reflection
symmetry of E. For a class of duality- symmetric Lagrangians we introduce an
alternative representation with the auxiliary scalar field and find new
explicit examples of such systems.
