We investigate the α+16O cluster structure in the inversion-doublet band (Kπ=01±) states of 20Ne with an angular-momentum-projected version of the Tohsaki-Horiuchi-Schuck-Röpke (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 single THSR wave functions, which are based on the concept of nonlocalized clustering, can well describe the Kπ=01- band and the Kπ=01+ 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 α+16O 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.