= Solution
The three coordinate <basis vectors> are <eigenvectors>, so $M$ is diagonal in those coordinates. Exactly one coordinate is multiplied by $-1$, while the other two are fixed. The three possibilities are
$$
\boxed{M=\operatorname{diag}(-1,1,1),\quad\operatorname{diag}(1,-1,1),\quad\operatorname{diag}(1,1,-1).}
$$
They are respectively reflections in the coordinate <planes> $x=0$, $y=0$ and $z=0$. Each is a <Householder reflection> with <unit normal> equal to the coordinate <eigenvector> carrying the <eigenvalue> $-1$.
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