Yes, the longitude of the Dehn surgery solid torus can go to any $(r, s)$ such that the 2x2 determinant $ps-qr = \pm 1$.  This because the various choices for the longitude all differ by homeomorphisms of the torus which extend to homeomorphisms of the solid torus.  

More generally, consider gluing together two manifolds $M$ and $N$ along a homeomorphism $f:\partial M \to \partial N$.  Let $h:\partial N\to \partial N$ be a homeomorphism which extends to a homeomorphism $h':N\to N$.  Then gluing via $f$ and gluing via $h\circ f$ yield homeomorphic manifolds.