  We define and illustrate the novel notion of dual integrable hierarchies, on
the example of the nonlinear Schr\"odinger (NLS) hierarchy. For each integrable
nonlinear evolution equation (NLEE) in the hierarchy, dual integrable
structures are characterized by the fact that the zero-curvature representation
of the NLEE can be realized by two Hamiltonian formulations stemming from two
distinct choices of the configuration space, yielding two inequivalent Poisson
structures on the corresponding phase space and two distinct Hamiltonians. This
is fundamentally different from the standard bi-Hamiltonian or generally
multitime structure. The first formulation chooses purely space-dependent
fields as configuration space; it yields the standard Poisson structure for
NLS. The other one is new: it chooses purely time-dependent fields as
configuration space and yields a different Poisson structure at each level of
the hierarchy. The corresponding NLEE becomes a {\it space} evolution equation.
We emphasize the role of the Lagrangian formulation as a unifying framework for
deriving both Poisson structures, using ideas from covariant field theory. One
of our main results is to show that the two matrices of the Lax pair satisfy
the same form of ultralocal Poisson algebra (up to a sign) characterized by an
$r$-matrix structure, whereas traditionally only one of them is involved in the
classical $r$-matrix method. We construct explicit dual hierarchies of
Hamiltonians, and Lax representations of the triggered dynamics, from the
monodromy matrices of either Lax matrix. An appealing procedure to build a
multi-dimensional lattice of Lax pair, through successive uses of the dual
Poisson structures, is briefly introduced.
