  We define a new lattice structure on the elements of a finite Coxeter group
W. This lattice, called the shard intersection order, is weaker than the weak
order and has the noncrossing partition lattice NC(W) as a sublattice. The new
construction of NC(W) yields a new proof that NC(W) is a lattice. The shard
intersection order is graded and its rank generating function is the W-Eulerian
polynomial. Many order-theoretic properties of the shard intersection order,
like Mobius number, number of maximal chains, etc., are exactly analogous to
the corresponding properties of NC(W). There is a natural dimension-preserving
bijection between simplices in the order complex of the shard intersection
order (i.e. chains in the shard intersection order) and simplices in a certain
pulling triangulation of the W-permutohedron. Restricting the bijection to the
order complex of NC(W) yields a bijection to simplices in a pulling
triangulation of the W-associahedron.
  The shard intersection order is defined indirectly via the polyhedral
geometry of the reflecting hyperplanes of W. Indeed, most of the results of the
paper are proven in the more general setting of simplicial hyperplane
arrangements.
