Suppose that $A$ is a reflection about a plane spanned by $u,v$ and let $w$ be orthogonal to this plane, then $Au=u$ and $Av=v$ since the reflection through the plane doesn't change them and $Aw = -w$ (this is the reflection).
Now switch the basis to $u,v,w$ and you will get the result.
If you have any two diagonal matrices, that have the same entries on their diagonal, but maybe in a different order, then they are similar.
You can see this by defining the matrix $E_\sigma$ for a permutation $\sigma \in S_n$ on n elements as $(E_\sigma)_{i,\sigma(i)}=1$ and all the other entries are zero. If $D$ is diagonal then $E_\sigma D E_\sigma ^{-1}$ is just the permutation of the entries on the diagonal.