  We investigate the anti-de Sitter (AdS) counterpart to the well studied de
Sitter (dS) model for energy-momentum space, viz "$\kappa$-momentum space"
space (with a structure based on the properties of the $\kappa$-Poincar\'e Hopf
algebra). On the basis of previous preliminary results one might expect the two
models to be "dual": dS exhibiting an invariant maximal spatial momentum but
unbounded energy, AdS a maximal energy but unbounded momentum. If that were the
case AdS momentum space could be used to implement a principle of maximal
Planck-scale energy, just as several studies use dS momentum space to postulate
of maximal Planck-scale spatial momentum. However several unexpected features
are uncovered in this paper, which limit the scope of the expected duality, and
interestingly they take different forms in different coordinatizations of AdS
momentum space. "Cosmological" AdS coordinates mimic the dS construction used
for $\kappa$-momentum space, and produce a Carrol limit in the ultraviolet.
However, unlike the $\kappa$-momentum space, the boundary of the covered patch
breaks Lorentz invariance, thereby introducing a preferred frame. In
"horospherical" coordinates we achieve full consistency with frame independence
as far as boost transformations are concerned, but find that rotational
symmetry is broken, leading to an anisotropic model for the speed of light.
Finally, in "static" coordinates we find a way of deforming relativistic
transformations that successfully enforces frame invariance and isotropy, and
produces a Carrol limit in the ultraviolet. However, the phenomenological
implications appear to be too weak for any realistic chance of detection. Our
results are also relevant for a long-standing debate on whether or not
coordinate redefinitions in momentum space lead to physically equivalent
theories: our three proposals are evidently physically inequivalent (abridged)
