First-principles electronic structure calculations within a gradient corrected density functional formalism have been carried out to investigate the electronic structure and magnetic properties of Pd-13 clusters. It is shown that a bilayer ground-state structure that can be regarded as a relaxed bulk fragment is most compatible with the experimental results from Stern-Gerlach measurements. An icosahedral structure, considered to be the ground state in numerous previous studies, is shown to be around 0.14 eV above the ground state. A detailed analysis of the molecular orbitals reveals the near degeneracy of the bilayer or icosahedral structures is rooted in the stabilization by p- or d-like cluster orbitals. The importance of low-lying spin states in controlling the electronic and magnetic properties of the cluster is highlighted.