  We consider the moduli space of rank 2 Higgs bundles with fixed determinant
over a smooth projective curve X of genus 2 over the complex numbers, and study
involutions defined by tensoring the vector bundle with an element $\alpha$ of
order 2 in the Jacobian of the curve, combined with multiplication of the Higgs
field by $\pm 1$. We describe the fixed points of these involutions in terms of
the Prym variety of the covering of $X$ defined by $\alpha$, and give an
interpretation in terms of the moduli space of representations of the
fundamental group.
