Can we parameterize Heegaard splittings?

Question

Let $M = V_1 \cup V_2$ be a 3-manifold with a Heegaard splitting. Then $V_1 \cap V_2 \cong \#^g S^1 \times S^1$ and $V_1, V_2 \cong \natural^g S^1 \times B^2$ (by definition) - but is it possible to choose homeomorphisms $V_1 \cong \natural^g S^1 \times B^2$ and $V_2 \cong \natural^g S^1 \times B^2$ such that the induced isomorphisms on $\#^g S^1 \times S^1 = \partial(\natural^g S^1 \times B^2)$ are the same?