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Assume that $M$ is a manifold.

Is there an embedding $M$ in some $\mathbb{R}^{n}$ such that the image of $M$ in $\mathbb{R}^{n}$ is invariant under each reflection $(x_{1},x_{2},\ldots x_{i},\ldots,x_{n}) \mapsto (x_{1},x_{2},\ldots , -x_{i},\ldots,x_{n})$, for all $i\in \{1,2,\ldots ,n\}$?

Assume that $M$ is a manifold.

Is there an embedding of $M$ in some $\mathbb{R}^{n}$ such that the image of $M$ in $\mathbb{R}^{n}$ is invariant under each reflection $(x_{1},x_{2},\ldots x_{i},\ldots,x_{n}) \mapsto (x_{1},x_{2},\ldots , -x_{i},\ldots,x_{n})$, for all $i\in \{1,2,\ldots ,n\}$?