  We construct $\sorth{p} \times \sorth{q}$-invariant special Lagrangian (SL)
cones in $\C^{p+q}$. These SL cones are natural higher-dimensional analogues of
the $\sorth{2}$-invariant SL cones constructed previously by MH and used in our
gluing constructions of higher genus SL cones in $\C^{3}$. We study in detail
the geometry of these $\sorth{p}\times \sorth{q}$-invariant SL cones, in
preparation for their application to our higher dimensional special Legendrian
gluing constructions. In particular the symmetries of these cones and their
asymptotics near the spherical limit are analysed. All $\sorth{p} \times
\sorth{q}$-invariant SL cones arise from a more general construction of
independent interest which we call the special Legendrian twisted product
construction. Using this twisted product construction and simple variants of it
we can construct a constellation of new special Lagrangian and Hamiltonian
stationary cones in $\C^{n}$. We prove the following theorems: A. there are
infinitely many topological types of special Lagrangian and Hamiltonian
stationary cones in $\C^{n}$ for all $n\ge 4$, B. for $n\ge 4$ special
Lagrangian and Hamiltonian stationary torus cones in $\C^{n}$ can occur in
continuous families of arbitrarily high dimension and C. for $n\ge 6$ there are
infinitely many topological types of special Lagrangian and Hamiltonian
stationary cones in $\C^{n}$ that can occur in continuous families of
arbitrarily high dimension.
