The reflection of infrared light by ionic crystals with cubic symmetry such as lithium fluoride, LiF, is analyzed in terms of phonon-polaritons. In contrast to the conventional view on phonon-polaritons that uses the Coulomb gauge and assumes a purely local dielectric response of the material, we here develop an alternative description making use of the Lorentz gauge. This involves retarded interactions between charges, implying a non-local response of the material to electromagnetic radiation. The resulting new phonon-polariton dispersion relation features polaritons with negative group velocity in the frequency range in between the transverse (ωT) and longitudinal frequency (ωL). By contrast, the conventional description predicts, in zero order, the absence of any propagating polaritons in the frequency interval between ωT and ωL. The new dispersion relation provides an efficient, zero-order description of the fine structure within the reststrahlen band of LiF. The local minimum near the middle of the reflectance band is due to excitation of a phonon-polariton whose energy and momentum matches that of the incoming photon. The Lorentz gauge description can also describe off-normal reflection and accounts for the experimentally observed widening of the reflection band with increasing angle of incidence.