  We study self-duality for interacting particle systems, where the particles
move as continuous time random walkers having either exclusion interaction or
inclusion interaction. We show that orthogonal self-dualities arise from
unitary symmetries of the Markov generator. For these symmetries we provide two
equivalent expressions that are related by the Baker-Campbell-Hausdorff
formula. The first expression is the exponential of an anti Hermitian operator
and thus is unitary by inspection; the second expression is factorized into
three terms and is proved to be unitary by using generating functions. The
factorized form is also obtained by using an independent approach based on
scalar products, which is a new method of independent interest that we
introduce to derive (bi)orthogonal duality functions from non-orthogonal
duality functions.
