  We show how to construct a graded locally compact Hausdorff \'etale groupoid
from a C*-algebra carrying a coaction of a discrete group, together with a
suitable abelian subalgebra. We call this groupoid the extended Weyl groupoid.
When the coaction is trivial and the subalgebra is Cartan, our groupoid agrees
with Renault's Weyl groupoid. We prove that if G is a second-countable locally
compact \'etale groupoid carrying a grading of a discrete group, and if the
interior of the trivially graded isotropy is abelian and torsion free, then the
extended Weyl groupoid of its reduced C*-algebra is isomorphic as a graded
groupoid to G. In particular, two such groupoids are isomorphic as graded
groupoids if and only if there is an equivariant diagonal-preserving
isomorphism of their reduced C*-algebras. We introduce graded equivalence of
groupoids, and establish that two graded groupoids in which the trivially
graded isotropy has torsion-free abelian interior are graded equivalent if and
only if there is an equivariant diagonal-preserving Morita equivalence between
their reduced C*-algebras. We use these results to establish rigidity results
for a number of classes of dynamical systems, including all actions of the
natural numbers by local homeomorphisms of locally compact Hausdorff spaces.
