= Solution
Choose the intersection point $p=[(\theta_0,\theta_1)]$ as the zero-cell. Let $a$ be the horizontal one-cell $S^1\times\{\theta_1\}$ and $b$ the vertical one-cell $\{\theta_0\}\times S^1$. The usual <CW complex> structure on the <torus> has one two-cell $c$ attached by the word $aba^{-1}b^{-1}$, and the extra disk gives a two-cell $d$ attached along $b$ with degree one. Collapsing $a$ leaves one zero-cell $p$, one one-cell $b$, and the two two-cells $c,d$. The <cellular boundary formula> gives
$$
C_2\cong\mathbb Zc\oplus\mathbb Zd,
\qquad
C_1\cong\mathbb Zb,
\qquad
\partial_2(c)=0,quad\partial_2(d)=b.
$$
Therefore
$$
H_i(X;\mathbb Z)\cong
\begin{cases}
\mathbb Z,&i=0,2,\\
0,&\text{otherwise}.
\end{cases}
$$
The <Excision theorem> says that if $Z\subseteq A\subseteq X$ and $\overline Z\subseteq\operatorname{int}A$, then inclusion induces
$$
H_i(X\setminus Z,A\setminus Z)\xrightarrow{\sim}H_i(X,A).
$$
It lets us compute <local homology> in arbitrarily small neighborhoods.
Let $C\subseteq X$ be the image of $\{\theta_0\}\times S^1$. At a point of $X\setminus C$, a neighborhood is a disk, whose link is a circle. At a point of $C\setminus\{p\}$, three half-disks meet along their diameters, and the link is a <Theta graph>, with first homology $\mathbb Z^2$. At $p$, the two folds created by collapsing the horizontal circle give a <dumbbell graph> as link: two circles joined by an interval. Its first homology is again $\mathbb Z^2$. The <local homology from a link> therefore gives
$$
H_i(X,X\setminus\{x\};\mathbb Z)
\cong
\begin{cases}
\mathbb Z^2,&i=2\text{ and }x\in C,\\
\mathbb Z,&i=2\text{ and }x\notin C,\\
0,&\text{otherwise}.
\end{cases}
$$
Because <local homology> is invariant under a <homeomorphism>, every self-homeomorphism of $X$ preserves the rank-two locus $C$.
It must also preserve $p$. Indeed, $p$ is the unique point of $C$ whose sufficiently small local link is a dumbbell graph; every other point of $C$ has a theta-graph link. The connecting edge of a dumbbell graph is a <bridge in a graph>, whereas no edge of a theta graph is a bridge, so these local topological types are distinct.
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