  Recently the collection $\cal G$ of all signed graphs for which the adjacency
matrix has all but at most two eigenvalues equal to $\pm 1$ has been
determined. Here we investigate $\cal G$ for cospectral pairs, and for signed
graphs determined by their spectrum (up to switching). If the order is at most
20, the outcome is presented in a clear table. If the spectrum is symmetric we
find all signed graphs in $\cal G$ determined by their spectrum, and we obtain
all signed graphs cospectral with the bipartite double of the complete graph.
In addition we determine all signed graphs cospectral with the Friendship graph
$F_\ell$, and show that there is no connected signed graph cospectral but not
switching equivalent with $F_\ell$.
