A result of Helgason's is that any self-intersecting geodesic in a Riemannian globally symmetric space is simple and closed. To what extent does this generalise to Riemannian naturally reductive homogeneous spaces with the canonical connection?
Every geodesic loop on a compact homogeneous space is a closed geodesic. Let c be such a loop c(L)=c(0). Take (n-1) Killing vector fields $X_i$ whose value at p are a basis of $c'(0)^\perp$. Then $X_i$ restricted to c is a Jacobi field and hence $<X_i(c(t),c'(t)>$ is linear. Since X has bounded length, $<X_i(c(t),c'(t)>=0$ and hence $c'(L)=c'(0)$. I think this may be also true if M is not compact.
For a homogeneous metric there is no reason for a geodesic loop at a point to be necessarily a closed geodesic. Counterexamples can probably be constructed already using metrics on the 3-sphere constructed by rescaling the round metric along the fiber of the Hopf fibration.