  In this paper we classify Legendrian and transverse knots in the knot types
obtained from positive torus knots by cabling. This classification allows us to
demonstrate several new phenomena. Specifically, we show there are knot types
that have non-destabilizable Legendrian representatives whose
Thurston-Bennequin invariant is arbitrarily far from maximal. We also exhibit
Legendrian knots requiring arbitrarily many stabilizations before they become
Legendrian isotopic. Similar new phenomena are observed for transverse knots.
To achieve these results we define and study "partially thickenable" tori,
which allow us to completely classify solid tori representing positive torus
knots.
