  In the traditional view, heavy deformed nuclei are like axially-symmetric
prolate ellipsoids, rotating about one of the short axes. In the present
picture, their shapes may be triaxial. The triaxial shape yields complex
rotations, which actually well reproduce experimental data, as confirmed by
state-of-the-art Configuration Interaction calculations. Two origins are
suggested for the triaxiality: (i) binding-energy gain by the symmetry
restoration for triaxial shapes, and (ii) another gain by specific components
of the nuclear force, like tensor force and high-multipole (e.g. hexadecupole)
central force. While the origin (i) produces basic modest triaxiality for
virtually all deformed nuclei, the origin (ii) produces more prominent
triaxiality for a certain class of nuclei. An example of the former is 154Sm, a
typical showcase of axial symmetry but is now suggested to depict a modest yet
finite triaxiality. The latter, prominent triaxiality, is discussed from
various viewpoints for some exemplified nuclei including 166Er, and
experimental findings. Many-body structures of the gamma band and the
double-gamma band are clarified. Regarding the general features of rotational
states of deformed many-body systems including triaxial ones, the well-known
J(J+1) rule of rotational excitation energies is derived, within the quantum
mechanical many-body theory, without resorting to the quantization of a
rotating classical rigid body. This derivation is extended to finite K. The
present picture of the rotation is robust and can be applied to various shapes
or configurations, including clusters and molecules. Thus, two long-standing
open problems, (i) occurrence and origins of triaxiality and (ii) quantum
many-body derivation of rotational energy, are resolved. Their possible
relations to Davydov's rigid-triaxial-rotor model are mentioned.
