A real orthogonal square matrix $Q$ of dimension $n$ is defined such that $Q Q^\dagger=I$.
The associated linear operation $x \in \mathbf{R}^n \rightarrow Qx \in \mathbf{R}^n$ has a geometric interpretation that can be visualized in two dimensions: for every vectors $u,v$, the scalar product $u \cdot v$ is equal to the scalar product between the images $Qu \cdot Qv$. This characterization is intuitive and thanks to it one can easily "visualize" how an orthogonal transformation will act on an object: it will transform it in a rigid way.
Now a real symmetric matrix is defined such that $A=A^\dagger$. We have the geometric characterization that the scalar product $Au \cdot v$ is equal to $Av \cdot u$, a condition which to me looks much less intuitive. Can one "visualize" a typical transformation associated to a symmetric matrix ?