  We investigate the $\alpha$+\oo\ cluster structure in the inversion-doublet
band ($K^\pi=0_{1}^\pm$) states of \nene\ with an angular-momentum-projected
version of the Tohsaki-Horiuchi-Schuck-R\"{o}pke (THSR) wave function, which
was successful "in its original 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.
