Assume that we have a vector bundle $E$ over $S^n$.

Is there a continuous family of invertible linear maps $T_x:E_x \to E_{-x}$?

Here continuity has the obvious meaning as soon as we have trivialization for the bundle around $x$ and $-x$

There is an obvious affirmative answer for the non orientable line bundle over $S^1$, for the tangent bundle of $S^n$ and for the $2$ dimensional real vector bundle $E \to S^2 \simeq \mathbb{C}P^1$ where $E$ is the tautological complex line bundle over $\mathbb{C}P^1$.

If the answer is yes, what about if we consider $E$ as a Riemannian bundle and we require that every $T_x$ be an isometric linear map?