Palladium (Pd) nanoparticles exhibit broad optical resonances that have been assigned to so-called localized surface plasmons (LSPs). The resonance's energy varies with particle shape in a similar fashion as is well known for LSPs in gold and silver nanoparticles, but the line-shape is always anomalously asymmetric. We here show that this effect is due to an intrinsic Fano interference caused by the coupling between the plasmon response and a structureless background originating from interband transitions. The conclusions are supported by experimental and numerical simulation data of Pd particles of different shape and phenomenologically analyzed in terms of the point dipole polarizability of spheroids. The latter analysis indicates that the degree of Fano asymmetry is simply linearly proportional to the imaginary part of the interband contribution to the metal dielectric function.