  We develop a general structure theory for compact homogeneous Riemannian
manifolds in relation to the co-index of symmetry. We will then use these
results to classify irreducible, simply connected, compact homogeneous
Riemannian manifolds whose co-index of symmetry is less or equal than three. We
will also construct many examples which arise from the theory of polars and
centrioles in Riemannian symmetric spaces of compact type.
