I want to study same 3-manifolds with different Heegaard splitings. Of course one has stabilization, but even with the same genus, we have different Heegaard splittings.

If we encode a 3-manifolds by a genus $g$, a set of curves $c_1,c_2,\ldots$ on the Heegaard surface of genus $g$, which are images of the meridian discs of one of the handlebodies, do we have an explicit set of "rules" (similar perhaps to Reidemeister moves) that tell us when two such encodings represent the same 3-manifold?