The electronic states of lateral many-electron quantum dots in high magnetic fields are analyzed in terms of energy and spin. In a regime with two Landau levels in the dot, several Coulomb-blockade peaks are measured. A zigzag pattern is found as it is known from the Fock-Darwin spectrum. However, only data from Landau level 0 show the typical spin-induced bimodality, whereas features from Landau level 1 cannot be explained with the Fock-Darwin picture. Instead, by including the interaction effects within spin-density-functional theory a good agreement between experiment and theory is obtained. The absence of bimodality on Landau level 1 is found to be due to strong spin polarization. © 2010 The American Physical Society.