  A useful concept in the development of physical models on the
$\kappa$-Minkowski noncommutative spacetime is that of a curved momentum space.
This structure is not unique: several inequivalent momentum space geometries
have been identified. Some are associated to a different assumption regarding
the signature of spacetime (i.e. Lorentzian vs. Euclidean), but there are
inequivalent momentum spaces that can be associated to the same signature and
even the same group of symmetries. Moreover, in the literature there are two
approaches to the definition of these momentum spaces, one based on the right-
(or left-)invariant metrics on the Lie group generated by the
$\kappa$-Minkowski algebra. The other is based on the construction of
$5$-dimensional matrix representation of the $\kappa$-Minkowski coordinate
algebra. Neither approach leads to a unique construction. Here, we find the
relation between these two approaches and introduce a unified approach, capable
of describing all momentum spaces, and identify the corresponding quantum group
of spacetime symmetries. We reproduce known results and get a few new ones. In
particular, we describe the three momentum spaces associated to the
$\kappa$-Poincar\'e group, which are half of a de Sitter, anti-de Sitter or
Minkowski space, and we identify what distinguishes them. Moreover, we find a
new momentum space with the geometry of a light cone, associated to a
$\kappa$-deformation of the Carroll group.
