We study two sorts of actions on the space of conjugacy classes of
irreducible $SU_2$-representations of a knot group. One of them is an
involution which comes from the algebraic structure of $SU_2$ and the other is
the action by the outer automorphism group of the knot group. In particular, we
consider them on an 1-dimensional smooth part of the space, which is
canonically oriented and metrized via a Reidemeister torsion volume form. As an
application we show that the Reidemeister torsion function on the 1-dimensional
subspace has symmetry about the metrization.