  A conference matrix of order $n$ is an $n\times n$ matrix $C$ with diagonal
entries $0$ and off-diagonal entries $\pm 1$ satisfying $CC^\top=(n-1)I$. If
$C$ is symmetric, then $C$ has a symmetric spectrum $\Sigma$ (that is,
$\Sigma=-\Sigma$) and eigenvalues $\pm\sqrt{n-1}$. We show that many principal
submatrices of $C$ also have symmetric spectrum, which leads to examples of
Seidel matrices of graphs (or, equivalently, adjacency matrices of complete
signed graphs) with a symmetric spectrum. In addition, we show that some Seidel
matrices with symmetric spectrum can be characterized by this construction.
