The underlying structure of low-lying collective bands of atomic nuclei is discussed from a novel perspective on the interplay between single-particle and collective degrees of freedom, by utilizing state-of-the-art configuration interaction calculations on heavy nuclei. Besides the multipole components of the nucleon-nucleon interaction that drive collective modes forming those bands, the monopole component is shown to control the resistance against such modes. The calculated structure of ^{154}Sm corresponds to the coexistence between prolate and triaxial shapes, while that of ^{166}Er exhibits a deformed shape with a strong triaxial instability. Both findings differ from traditional views based on β/γ vibrations. The formation of collective bands is shown to be facilitated from a self-organization mechanism.