  We investigate properties of signed graphs that have few distinct eigenvalues
together with a symmetric spectrum. Our main contribution is to determine all
signed $(0,2)$-graphs with vertex degree at most $6$ that have precisely two
distinct eigenvalues $\pm \lambda$. Next, we consider to what extent induced
subgraphs of signed graph with two distinct eigenvalues $\pm \lambda$ are
determined by their spectra. Lastly, we classify signed $(0,2)$-graphs that
have a symmetric spectrum with three distinct eigenvalues and give a partial
classification for those with four distinct eigenvalues.
