  In this paper we derive intertwining relations for a broad class of
conservative particle systems both in discrete and continuous setting. Using
the language of point process theory, we are able to derive a natural framework
in which duality and intertwining can be formulated. We prove falling factorial
and orthogonal polynomial intertwining relations in a general setting. These
intertwinings unite the previously found classical and orthogonal
self-dualities in the context of discrete particle systems and provide new
dualities for several interacting systems in the continuum. We also introduce a
new process, the symmetric inclusion process in the continuum, for which our
general method applies and yields generalized Meixner polynomials as orthogonal
self-intertwiners.
