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?
 A: 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: $A$ 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 $-1$.
A: 
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  $.
A: You can construct orthogonal and symmetric matrices using a nice parametrization from Sanyal [1] and Mortari [2].
We want a matrix $R$ both orthogonal and symmetric, i.e.
$$
R^T R = I \ \ \text{and}\ \  R = R^T
$$
which also means $R^2 = I$ and arbitrary powers are well-behaved
$$
R^k = \begin{cases}
R \ \ \ \text{if} \ \ k \ \ \ \text{is odd} \\
I \ \ \ \text{if} \ \ k \ \ \ \text{is even}
\end{cases}
$$
And now the parametrization.  One can construct such a matrix with a choice of $n$ orthogonal vectors $\{r_k\}_{k=1}^n$ and the desired number of positive eigenvalues $p \in [0,n]$
$$
R = \sum_{k=1}^p r_kr_k^T - \sum_{k=p+1}^n r_kr_k^T
$$
They also point out that

if $p=n$, then $R=I$ whereas if $p=0$, then $R=-I$.



*

*Sanyal, A. K., Geometrical Transformations in Higher Dimensional Euclidean
Spaces, Master’s thesis, Department of Aerospace Engineering, Texas A&M University,
College Station, TX, May 2001.

*Mortari, D.. (2004). Ortho-Skew and Ortho-Sym Matrix Trigonometry. Advances in the Astronautical Sciences. 115.

