Is this true for any rational homology 3-sphere Y (or any 3-manifold where $$K \subset Y$$ is null-homologous)? $$Y_{\frac{p}{q}} (K) = (-Y)_{- \frac{p}{q}} (m(K))$$ where m(K) is the mirror of K in -Y.
Any reference or a sketch of proof will be real helpful.
For an integral homology 3-sphere, your relation is very nearly the proof of Thm. 2.8.v, p. 198 in Boyer and Lines, regarding properties of their extension of Casson's invariant to homology lens spaces. Walker extended Casson's invariant (differently) to rational homology 3-spheres. I'm not sure how surprised I would be if a similar argument worked for this other invariant.
Boyer, Steven and Daniel Lines, Surgery formulae for Casson's invariant and extensions to homology lens spaces. J. reine angew. Math. 405 (1990), 181-220. https://eudml.org/doc/153216
Walker, K., An extension of Casson's invariant. Annals of Mathematics Studies, 126. Princeton University Press, Princeton, NJ, 1992. ISBN 0-691-08766-0, ISBN 0-691-02532-0