  The theta characteristics on a Riemann surface are permuted by the induced
action of the automorphism group, with the orbit structure being important for
the geometry of the curve and associated manifolds. We describe two new methods
for advancing the understanding of these orbits, generalising existing results
of Kallel & Sjerve, allowing us to establish the existence of infinitely many
curves possessing a unique invariant characteristic as well as determine the
number of invariant characteristics for all Hurwitz curves with simple
automorphism group. In addition, we compute orbit decompositions for a
substantial number of curves with genus $\leq 9$, allowing the identification
of where current theoretical understanding falls short and the potential
applications of machine learning techniques.
