 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 areThey 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 $-1$.