= Solution
Since $a^2=I$, a <representation> satisfies $D(a)^2=I_2$. Orthogonality gives $D(a)^{-1}=D(a)^T$, while the involution relation gives $D(a)^{-1}=D(a)$. Hence $D(a)$ is a <symmetric matrix>, so $\gamma=\delta$.
Its entries obey
$$
\alpha^2+\gamma^2=1,\qquad\beta^2+\gamma^2=1,\qquad\gamma(\alpha+\beta)=0.
$$
If $\gamma\ne0$, then $\beta=-\alpha$; putting $\alpha=\cos\theta$ and $\gamma=\sin\theta$ gives a reflection. If $\gamma=0$, the diagonal entries are independently $\pm1$: the opposite-sign cases are already included among these reflections, while equal signs give $I_2$ or $-I_2$. Faithfulness excludes $I_2$, since $a\ne I$. Therefore the <orthogonal involutions of the plane> give
$$
\boxed{D(a)=-I_2\quad\text{or}\quad D(a)=\begin{pmatrix}\cos\theta&\sin\theta\\\sin\theta&-\cos\theta\end{pmatrix}.}
$$
The first is rotation by $\pi$. The second is reflection in the line making angle $\theta/2$ with the first axis: the unit vector along that line is fixed, and its perpendicular is negated. These are all the nonidentity orthogonal transformations of order two in the plane.
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