We investigate the $alpha$+oo cluster structure in the inversion-doublet band ($K^pi=0_{1}^pm$) states of ene with an angular-momentum-projected version of the Tohsaki-Horiuchi-Schuck-R{o}pke (THSR) wave function, which was successful in its origina
l form for the description of, e.g., the famous Hoyle state. In contrast with the traditional view on clusters as localized objects, especially in inversion doublets, we find that these {it single} THSR wave functions, which are based on the concept of nonlocalized clustering, can well describe the $K^{pi}=0_1^-$ band and the $K^{pi}=0_1^+$ band. For instance, they have 99.98% and 99.87% squared overlaps for $1^-$ and $3^- $ states (99.29%, 98.79% and 97.75% for $0^+, 2^+$ and $4^+$ states), respectively, with the corresponding exact solution of the $alpha$+oo resonating group method. These astounding results shed a completely new light on the physics of low energy nuclear cluster states in nuclei: The clusters are nonlocalized and move around in the whole nuclear volume, only avoiding mutual overlap due to the Pauli blocking effect.
