  Quantum homogeneous spaces are noncommutative spaces with quantum group
covariance. Their semiclassical counterparts are Poisson homogeneous spaces,
which are quotient manifolds of Lie groups $M=G/H$ equipped with an additional
Poisson structure $\pi$ which is compatible with a Poisson-Lie structure $\Pi$
on $G$. Since the infinitesimal version of $\Pi$ defines a unique Lie bialgebra
structure $\delta$ on the Lie algebra $\frak g=\mbox{Lie}(G)$, we exploit the
idea of Lie bialgebra duality in order to study the notion of complementary
dual homogeneous space $M^\perp=G^\ast/H^\perp$ of a given homogeneous space
$M$ with respect to a coisotropic Lie bialgebra. Then, by considering the
natural notions of reductive and symmetric homogeneous spaces, we extend these
concepts to $M^\perp$ thus showing that an even richer duality framework
between $M$ and $M^\perp$ arises from them. In order to analyse physical
implications of these notions, the case of $M$ being a Minkowski or (Anti-) de
Sitter Poisson homogeneous spacetime is fully studied, and the corresponding
complementary dual reductive and symmetric spaces $M^\perp$ are explicitly
constructed in the case of the well-known $\kappa$-deformation, where the
cosmological constant $\Lambda$ is introduced as an explicit parameter in order
to describe all Lorentzian spaces simultaneously. In particular, the fact that
$M^\perp$ is a reductive space is shown to provide a natural condition for the
representation theory of the quantum analogue of $M$ that ensures the existence
of physically meaningful uncertainty relations between the noncommutative
spacetime coordinates. Finally, despite these dual spaces $M^\perp$ are not
endowed in general with a $G^\ast$-invariant metric, we show that their
geometry can be described by making use of $K$-structures.
