Geometric interpretation of a symmetric matrix

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 ?