  We study Legendrian and transverse realizations of the negative torus knots
$T_{(p,-q)}$ in all contact structures on the $3$-sphere. We give a complete
classification of the strongly non-loose transverse realizations and the
strongly non-loose Legendrian realizations with the Thurston-Bennequin
invariant smaller than $-pq$.
  Additionally, we show that the strongly non-loose transverse realizations $T$
are classified by their non-zero invariants $\mathfrak T(T)$ in the minus
version of the knot Floer homology. However, not all the elements of
$HFK^-(T_{(p,q)})$ can be realized.
  Along the way, we relate our Legendrian realizations to the tight contact
structures on the Legendrian surgeries along them. Specifically, we realize all
tight structures on the lens spaces $L(pq+1,p^2)$ as a single Legendrian
surgery on a Legendrian $T_{(p,-q)}$, and we relate transverse realizations in
overtwisted structures to the non-fillable tight structures on the large
negative surgeries along the underlying knots.
