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For a knot $K$ with a strong inversion $i$ induced by an unknotting tunnel, we have a double covering projection $\mathrm{\Pi }:S^3\to S^3/i$ branched over a trivial knot $\mathrm{\Pi }\left(\text{fix}\left(i\right)\right)$, where $\text{fix}\left(i\right)$ is the axis of $i$. Then a set $\mathrm{\Pi }\left(\text{fix}\left(i\right)\cup K\right)$ is called a $\theta$-curve. We construct $\theta$-curves and the ${\mathbb{Z}}_2\oplus {\mathbb{Z}}_2$ cyclic branched coverings over $\theta$-curves, having two non-isotopic Heegaard decompositions which are one stable equivalent.