A symmetric matrix has orthogonal eigenvectors with real eigenvalues, and hence can be thought of as scaling along a particular orthogonal set of axes.
Of course, not all vectors are (usually) eigenvectors, and therefore many individual vectors will have their direction changed under a symmetric matrix.
It seems to me, however, that these direction changes will somehow "cancel out" in the sense that there is no net rotation by a symmetric matrix. I don't know how to state this more precisely, other than to observe that $A_{ij} - A_{ji}$ is a measure of the rotation in the $i,j$ plane (see here).
That is, if $A$ is a symmetric matrix, for any plane $P$, the net direction change of $Av$ over all vectors $v \in P$ is zero. Again, I'm not sure how state this more precisely, but perhaps it could be formulated as an integral.
Is my intuition correct? And how can it be stated more precisely?