What can be said about a matrix which is both symmetric and orthogonal?

I tried to find matrices A, which are both orthogonal and symmetric, this means
$A=A^{-1}=A^T$. I only found very special examples like I, -I or the matrix
$$\begin{pmatrix}
0 &0& -1\\
0& -1& 0\\
-1& 0& 0
\end{pmatrix} $$
Can a matrix with the desired properties only contain the values -1,0 and 1 ?
Which matrices of a given size have the desired property ?

Solutions Collecting From Web of "What can be said about a matrix which is both symmetric and orthogonal?"

For your first question, the answer is no. Every real Householder reflection matrix is a symmetric orthogonal matrix, but its entries can be quite arbitrary.

In general, if $A$ is symmetric, it is orthogonally diagonalisable and all its eigenvalues are real. If it is also orthogonal, its eigenvalues must be 1 or -1. It follows that every symmetric orthogonal matrix is of the form $QDQ^\top$, where $Q$ is a real orthogonal matrix and $D$ is a diagonal matrix whose diagonal entries are 1 or -1.

$A$ is orthogonal and symmetric, so $A=A^{-1}$ and $A=A^{T}$. More
general, let $A$ be a unitary and self-adjoint operator with discrete
spectrum in a separable Hilbert space. Then $A=\exp [iW]$ with $W$
self-adjoint and $A=A^{\ast }=\exp [-iW]$. Thus $W=\sum_{n}\lambda _{n}P_{n}$
with $\lambda _{n}\in \mathbb{R}$ and the $P_{n}$ are orthogonal projectors,
$\lambda _{m}\neq \lambda _{n}$, $m\neq n$ and $P_{m}P_{n}=\delta _{mn}P_{m}$
. Now
\begin{equation*}
A=\sum_{n}\exp [i\lambda _{n}]P_{n}=A^{\ast }=\sum_{n}\exp [-i\lambda
_{n}]P_{n},
\end{equation*}
so $\exp [2i\lambda _{n}]=1$ leading to $\lambda _{n}=k_{n}\pi $, $k_{n}\in
\mathbb{Z}$, which is either $+1$ or $-1$.

Can a matrix with the desired properties only contain the values -1,0 and 1 ?

For this part of your question every 3-D rotation matrix (it’s orthogonal) about any axis ( defined by a unit vector $v$) by angle $\pi$ is symmetric.

You can generate plenty of them with Rodrigues’ rotation formula which for a $\pi$ case takes simpler form $rot(v, \pi)= 2vv^T-I$ and they are not necessary consist only of $-1, 0, 1 $.