  In this work we present and analyze a new class of coordinate representations
of de Sitter and anti-de Sitter spacetimes for which the metrics are diagonal
and (typically) static and axially symmetric. Contrary to the well-known forms
of these fundamental geometries, that usually correspond to a 1+3 foliation
with the 3-space of a constant spatial curvature, the new metrics are adapted
to a 2+2 foliation, and are warped products of two 2-spaces of constant
curvature. This new class of (anti-)de Sitter metrics depends on the value of
cosmological constant Lambda and two discrete parameters +1,0,-1 related to the
curvature of the 2-spaces. The class admits 3 distinct subcases for Lambda>0
and 8 subcases for Lambda<0. We systematically study all these possibilities.
In particular, we explicitly present the corresponding parametrizations of the
(anti-)de Sitter hyperboloid, visualize the coordinate lines and surfaces
within the global conformal cylinder, investigate their mutual relations,
present some closely related forms of the metrics, and give transformations to
standard de Sitter and anti-de Sitter metrics. Using these results, we also
provide a physical interpretation of B-metrics as exact gravitational fields of
a tachyon.
