  A signed graph is said to be sign-symmetric if it is switching isomorphic to
its negation. Bipartite signed graphs are trivially sign-symmetric. We give new
constructions of non-bipartite sign-symmetric signed graphs. Sign-symmetric
signed graphs have a symmetric spectrum but not the other way around. We
present constructions of signed graphs with symmetric spectra which are not
sign-symmetric. This, in particular answers a problem posed by Belardo,
Cioab\u{a}, Koolen, and Wang (2018).
