A beta-plane approximation of the two-dimensional quasigeostrophic model describes a single layer (barotropic) fluid subjected to a latitudinally varying Coriolis parameter or topography. Rhines (1975) initiated the study of beta-plane turbulence. He predicted the inverse energy cascade into predominantly zonal modes, hence an array of eastward-westward jets, and estimated the jet number (celebrated Rhines scale). He also proposed a k-5 scaling law of zonal energy spectra. Our paper re-examines scaling, spectra, and zonal structure of beta-plane turbulence, based on theoretical predictions and numeric experiments. We show that the inverse cascade gives rise to strong organized zonal jets that evolve a peculiar frontal-band ("saw-tooth") vorticity profile. Such structure affects all spectral properties of the system, by creating organized sequences of spectral peaks, and thus confounds any putative "scaling behavior." The frontal-band structure appears consistently in all stochastically forced beta-plane flows, independent of dissipation and/or other details. But the resulting turbulent quasiequilibrium is not unique, its gross parameters (jets number, mean vorticity gradient) retain memory of the initial state and/or history. © 2004 American Institute of Physics.