  We establish duality between real forms of the quantum deformation of the
4-dimensional orthogonal group studied by Fioresi et al. and the classification
work made by Borowiec et al.. Classically these real forms are the isometry
groups of $\mathbb{R}^4$ equipped with Euclidean, Kleinian or Lorentzian
metric. A general deformation, named $q$-linked, of each of these spaces is
then constructed, together with the coaction of the corresponding isometry
group.
