Question about existence of special function Is there any orientation-preserving homeomorphism $g:\mathbb{R}^n\rightarrow\mathbb{R}^n$ s.t. $\forall x\in\mathbb{R}^n,g(g(x))=x$ that is not $g(x)=x$ or $g(x)=-x$?

Yes. For instance, take $f\colon\mathbb{R}^3\longrightarrow\mathbb{R}^3$ defined by $f(x,y,z)=(-x,-y,z)$.
