Using the spin-polarized method it is shown that the periodicity of the various properties of lanthanides and actinides is related with the "electron-vacancy" symmetry for the ground state of the f(N) shell. This symmetry remains approximately valid also taking into account some important terms and configuration mixing effects. The algebraic expressions for the ground-state energy and the ionization energies in terms of the number of f electrons are obtained and the interval rule is derived. The reason for the inclined W systematic or L dependence is the correlation between the values of the two quantities, depending almost linearly on the number of f electrons in the four quarters of this shell.