  The Goeritz group of a genus $g$ Heegaard splitting of a 3-manifold is the
group of isotopy classes of orientation-preserving automorphisms of the
manifold that preserve the Heegaard splitting. In the context of the standard
genus 2 Heegaard splitting of $S^3$, we introduce the concept of Goeritz
equivalence of curves, present two algebraic obstructions to Goeritz
equivalence of simple closed curves that are straightforward to compute, and
provide families of examples demonstrating how these obstructions may be used.
