Call a subset  $Z$ of $\mathbb{S}^n$ *ambiently-reversible*, if there is an orientation-reversing automorphism $h: \mathbb{S}^n \to \mathbb{S}^n$ fixing $Z$ pointwise.


**Question 1:** Which subsets of $\mathbb{S}^n, n \geq 2$ are ambiently-reversible?

This may be hard, so let me ask a more specific question: 

**Question 2:** Let $Z \subset \mathbb{S}^2$ be closed. Is it true that $Z$ is ambiently-reversible iff it is homeomorphic with a subspace of $\mathbb{S}^1$?

Note that if $Z$ is contained in a homeomorph $K$ of $\mathbb{S}^1$ where $K\subset \mathbb{S}^2$, then $Z$ is ambiently-reversible. To see this, apply Jordan-Shoenflies to $K$, and reflect its two sides. This can be used, in combination with the Denjoy–Riesz theorem, to show that every closed 0-dimensional subset of $\mathbb{S}^2$ is ambiently-reversible.

Remark: If $Z$ is ambiently-reversible, then so is its closure.

See also [this question][1].


  [1]: https://mathoverflow.net/questions/428999/subsets-of-mathbbsn-fixed-by-an-orientation-reversing-automorphism-part