In materials with nearly commensurate band filling the electron liquid may spontaneously separate into components with distinct properties, yielding complex intra- and interunit cell ordering patterns and a reduced dimensionality. Polarization-dependent angle-resolved photoemission data with submicron spatial resolution demonstrate such an electronic self-organization in NbSe3, a compound considered to be a paradigm of charge order. The new data indicate the emergence of a novel order, and reveal the one-dimensional (1D) physics hidden in a material which naively could be considered the most three dimensional of all columnar chalcogenides. The 1D physics is evidenced by a new selection rule-in two polarizations we observe two strikingly different dispersions each closely resembling apparently contradicting results of previous studies of this material.