The symmetry underlying the geometric structure of materials determines most of their physical properties. In low dimensional systems the role of symmetry is enhanced and can give rise to new phenomena. Here, we report on unexpected magnetism in carbon nanotubes and O-rich surfaces of ionic oxides, to show how its existence is closely related to the symmetry conditions. First, based on tight-binding models, we demonstrate that chiral carbon nanotubes present spin splitting at the Fermi level in the absence of a magneticfield, whereas achiral tubes preserve spin degeneracy. These remarkably different behaviors of chiral and non-chiral nanotubes are due to the intrinsic symmetry dependence of the spin-orbit interaction. Second, the occurence of spin-polarization at ZrO2, Al2O3 and MgO surfaces is proved by means of abinitio calculations within the density functional theory. Large spin moments develop at O-ended polar terminations, transforming the non-magnetic insulator into a half-metal. The magnetic moments mainly reside in the surface oxygen atoms, and their origin is related to the existence of 2p holes of well-defined spin polarization at the valence band of the ionic oxide. The direct relation between magnetization and local loss of donor charge shows that at the origin of these phenomena is the reduced surface symmetry.