  We explore in detail the role in euclidean 3d quantum gravity of quantum Born
reciprocity or `semidualisation'. The latter is an algebraic operation defined
using quantum group methods that interchanges position and momentum. Using this
we are able to clarify the structural relationships between the effective
non-commutative geometries that have been discussed in the context of 3d
gravity. We show that the spin model based on D(U(su_2)) for quantum gravity
without cosmological constant is the semidual of a quantum particle on a
three-sphere, while the bicrossproduct (DSR) model is the semidual of a quantum
particle on hyperbolic space. We show further how the different models are all
specific limits of q-deformed models with q=e^{-\hbar \sqrt{-\Lambda}/m_p},
where m_p is the Planck mass and \Lambda is the cosmological constant, and
argue that semidualisation interchanges m_p and l_c, where l_c is the
cosmological length scale l_c=1/\sqrt{|\Lambda|}. We investigate the physics of
semidualisation by studying representation theory. In both the spin model and
its semidual we show that irreducible representations have a physical picture
as solutions of a respectively non-commutative/curved wave equation. We
explain, moreover, that the q-deformed model, at a certain algebraic level, is
self-dual under semidualisation.
