This paper is devoted to characterizing the so-called order isomorphisms intertwining the L2 -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.
MSC Classification: 60J46 , 60J35 , 31C25