Is $f(x)=(I-\frac{Wxx^TW^T}{\|Wx\|^2})x$ invertible on n-sphere if Jacobian is always non-zero?

Question

Let $f(x)=\left(I-2\frac{Wxx^TW^T}{\|Wx\|^2}\right)x$ for $x\in \mathbb{R}^n$ and $W\in\mathbb{R}^{n\times n}$.

Suppose I could constraint $W$ so the Jacobian determinant $\det \left(\frac{\partial f(x)}{\partial x}\right)\neq 0$ for all $x$ in the n-sphere $S^{n-1}=\{x\in\mathbb{R}^{n} \mid \|x\|=1\}$.

Is $f$ then invertible on $S^d$? If, can we compute or approximate the inverse somehow?