We analyze several puzzling features of a recent experiment with a noninteracting gas of atoms in a quadrupole trap. After an initial momentum kick, the system reaches a stationary, quasithermal state even without collisions, due to the dephasing of individual particle trajectories. Surprisingly, the momentum distribution remains anisotropic at long times, characterized by different temperatures along the different directions. In particular, there is no transfer of the kick energy between the axial and radial trap directions. To understand these effects we discuss and solve two closely related models: a spherically symmetric trap V(r)≃rα and a strongly confined gas along one direction (a pancake trap). We find that in the isotropic trap, the gas unexpectedly also preserves the anisotropy of the kick at long times, which we are able to explain using the conservation of angular momentum and the virial theorem. Depending on the value of α we find that the kick can cool or heat the orthogonal directions. The pancake trap case is quantitatively similar to the quadrupole one. We show that for the former, the temperature anisotropy and memory of the kick direction are due to the change in the two-dimensional effective potential resulting from the kick, thereby also explaining the quadrupole experimental results.