  In this paper, we present extraordinary algebraic and geometrical structures
for the Hunter-Saxton equation: infinitely many commuting and non-commuting
$x,t$-independent higher order symmetries and conserved densities. Using a
recursive relation, we explicitly generate infinitely many higher order
conserved densities dependent on arbitrary parameters. We find three Nijenhuis
recursion operators resulting from Hamiltonian pairs, of which two are new.
They generate three hierarchies of commuting local symmetries. Finally, we give
a local recursion operator depending on an arbitrary parameter.
  As a by-product, we classify all anti-symmetric operators of a definite form
that are compatible with the Hamiltonian operator $D_x^{-1}$.
