In this article, we classify the set of asymptotic mass-like invariants for asymptotically hyperbolic metrics. It turns out that the standard mass is just one example (but probably the most important one) among the two families of invariants we find. These invariants are attached to finite-dimensional representations of the group of isometries of hyperbolic space. They are then described in terms of wave harmonic polynomials and polynomial solutions to the linearized Einstein equations in Minkowski space.