  It is shown that the isometry group of the de Sitter spacetime includes two
different three-dimensional Abelian subgroups which transform between
themselves through a discrete isometry corresponding to the time reversal in
the five-dimensional Minkowski spacetime embedding the de Sitter one. The
eigenfunctions of the generators of these Abelian subgroups form two different
sets of quantum modes correlated by the mentioned isometry.
