  This paper is devoted to characterizing the so-called order isomorphisms
intertwining the $L^2$-semigroups of two Dirichlet forms. We first show that
every unitary order isomorphism intertwining semigroups is the composition of
$h$-transformation and quasi-homeomorphism. In addition, under the absolute
continuity condition on Dirichlet forms, every (not necessarily unitary) order
isomorphism intertwining semigroups is the composition of $h$-transformation,
quasi-homeomorphism, and multiplication by a certain step function.
