A symmetric space is a Riemannian manifold M with the following property: For every point $p \in M$ there is an isometry $\phi: M \rightarrow M$ such that $\phi(p) = p$ and $\phi_*(v_p) = -v_p \in T_pM$.
I am interested in the following property of a Riemannian manifold $M$: For every $p \in M$ and $H \subset T_pM$ a subspace there is an isometry $\phi: M \rightarrow M$ fixing p such that $\phi_*: T_pM \rightarrow T_pM$ is the reflection through the hyperplane $H$.
Has every symmetric space the above property? (The two dimensional simply connected symmetric spaces seem to have it)
A flat torus (that is, a rectangle with opposite sides identified) would seem to be a counterexample. It is symmetric, but for a generic point there are only a few directions you can reflect across.
On the torus, geodesics from a chosen point in different directions may either meet it again or not, so there cannot be an isometry that takes an arbitrary geodesic through the point to an arbitrary other one. However for any two rays in $\mathbb R^2$ there's always a line such that reflecting in it takes one to the other.
Basically you're requiring isotropy (because any rotation of the tangent space can be made as a composition of reflections), which severely limits the manifolds that work. I think all you'll get are spheres, Euclidean spaces, and hyperbolic ones (at various scales).