Consider the symmetric matrices associated to orthogonal reflections of the Euclidean plane.
Since a line comes back to itself after a half-turn, you get a family indexed by $[0,\pi]$
describing a closed path in the set of symmetric matrices. A corresponding eigenvector of eigenvalue $1$ and norm $1$, deformed continuously goes into its opposite. This path 
can thus not be lifted into a continuous part formed by eigenvectors.