  We study the symmetries of the spectrum of the Feinberg-Zee Random Hopping
Matrix. Chandler-Wilde and Davies proved that the spectrum of the Feinberg-Zee
Random Hopping Matrix is invariant under taking square roots, which implied
that the unit disk is contained in the spectrum (a result already obtained
slightly earlier by Chandler-Wilde, Chonchaiya and Lindner). In a similar
approach we show that there is an infinite sequence of symmetries at least in
the periodic part of the spectrum (which is conjectured to be dense). Using
these symmetries, we can exploit a considerably larger part of the spectrum
than the unit disk. As a further consequence we find an infinite sequence of
Julia sets contained in the spectrum. These facts may serve as a part of an
explanation of the seemingly fractal-like behaviour of the boundary.
