  Let $f$ and $g$ be two circle endomorphisms of degree $d\geq 2$ such that
each has bounded geometry, preserves the Lebesgue measure, and fixes $1$. Let
$h$ fixing $1$ be the topological conjugacy from $f$ to $g$. That is, $h\circ
f=g\circ h$. We prove that $h$ is a symmetric circle homeomorphism if and only
if $h=Id$. Many other rigidity results in circle dynamics follow from this very
general symmetric rigidity result.
