differential geometry – Is inversion globally well defined on complete anaytic manifods

Let $M$ be a complete analytic Riemannian manifold. Let $p$ be a point in $M$ and let $u,v$ be tangents at $p$ such that $exp(u)=exp(v)$, must we have $exp(-u)=exp(-v)$ ?

This property seems to good to be true, so I would not be surprised if there is a counterexample ,on the other hand I dont know any counterexamples.