Suppose we have a 3-manifold $M$ obtained by Dehn surgery along a given framed link on $S^3$. Then it has a natural orientation which comes from the standard orientation of $S^3$. Is it true that $-M$ ($M$ with orientation reversed) is obtained by Dehn surgery along the same link with sign-reversed coefficients on $S^3$?
I only know that this is true for one special case; $L(p,q)$ is obtained by $-p/q$-surgery along the unknot, $-L(p,q)=L(p,p-q)=L(p,-q)$ is obtained by $p/q$-surgery along the unknot. I want to know that is this true in general.
P.S. Is this true in the special case that $M$ is obtained by Dehn surgery along a single knot in $S^3$?
The answer is yes if you add a mirror reflection to the link (see Statement below)! It can be found in Saveliev Chapter 3.4 and see the doctoral dissertation Müllner for associated questions.
Statement: If $M$ is obtained by surgery along the framed link $L \subset \mathbb{S}^3$, then the reversed orientated $-M$ is obtained by surgery along $L^\ast$ which denotes the mirror image of $L$ along (any) hyperplane in $\mathbb{R}^3$ and sign-opposite coefficients to each link component.
Argument:
Reflecting the link complement along a plane will 1) reverse orientation, 2) mirror the link and 3) change the sign of the coefficients (as it reverses orientations of meridians and parallels).
Remark: Saveliev considers integral surgery in particular, but the same argument applies for rational surgery.