  We study pairs of Dirichlet forms related by an intertwining order
isomorphisms between the associated $L^2$-spaces. We consider the measurable,
the topological and the geometric setting respectively. In the measurable
setting, we deal with arbitrary (irreducible) Dirichlet forms and show that any
intertwining order isomorphism is necessarily unitary (up to a constant). In
the topological setting we deal with quasi-regular forms and show that any
intertwining order isomorphism induces a quasi-homeomorphism between the
underlying spaces. In the geometric setting we deal with both regular Dirichlet
forms as well as resistance forms and essentially show that the geometry
defined by these forms is preserved by intertwining order isomorphisms. In
particular, we prove in the strongly local regular case that intertwining order
isomorphisms induce isometries with respect to the intrinsic metrics between
the underlying spaces under fairly mild assumptions. This applies to a wide
variety of metric measure spaces including $\mathrm{RCD}(K,N)$-spaces, complete
weighted Riemannian manifolds and complete quantum graphs. In the non-local
regular case our results cover in particular graphs as well as fractional
Laplacians as arising in the treatment of $\alpha$-stable L\'evy processes. For
resistance forms we show that intertwining order isomorphisms are isometries
with respect to the resistance metrics.
  Our results can can be understood as saying that diffusion always determines
the Hilbert space, and -- under natural compatibility assumptions -- the
topology and the geometry respectively. As special instances they cover earlier
results for manifolds and graphs.
