We present ab initio study of the interplay between atomic structure and magnetic properties in palladium atomic-size contacts suspended between palladium electrodes. This study was motivated by recent controversy between Delin et al. (Phys. Rev. Lett. 96, 079702 (2006)) and Alexandre et al. (Phys. Rev. Lett. 96, 079701 (2006)) about possible onset of magnetism in palladium (Pd) contacts. Here, we elucidate the effect of atomic arrangement and structure relaxations revealed in contracted and elongated Pd contacts on their magnetic properties. Planelike and pyramidlike geometries for electrodes are studied. We demonstrate that palladium contacts exhibit magnetic properties distributed inhomogeneously within the contact for a wide range of distances between separated electrodes.