  We introduce a geometric invariant that we call the index of symmetry, which
measures how far is a Riemannian manifold from being a symmetric space. We
compute, in a geometric way, the index of symmetry of compact naturally
reductive spaces. In this case, the so-called leaf of symmetry turns out to be
of the group type. We also study several examples where the leaf of symmetry is
not of the group type. Interesting examples arise from the unit tangent bundle
of the sphere of curvature 2, and two metrics in an Aloff-Wallach 7-manifold
and the Wallach 24-manifold.
