We investigate the anti-de Sitter (AdS) counterpart to the well-studied de Sitter (dS) model for energy-momentum space, viz ``$\ensuremath{\kappa}$-momentum space'' space (with a structure based on the properties of the $\ensuremath{\kappa}$-Poincar\'e Hopf algebra). On the basis of previous preliminary results one might expect the two models to be complementary: 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 complementarity, and interestingly they take different forms in different coordinatizations of AdS momentum space. ``Cosmological'' AdS coordinates mimic the dS construction used for $\ensuremath{\kappa}$-momentum space, and produce a Carrol limit in the ultraviolet. However, unlike the $\ensuremath{\kappa}$-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. 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, leading to alternative models of Planck-scale effects. As a corollary we study the UV running of the Hausdorff dimension of momentum space in the first and third model, obtaining different results.