A beta-plane approximation of the 2D quasigeostrophic modeldescribes a single layer (barotropic) fluid subjected to alatitudinally varying Coriolis parameter or topography. Rhines 1975initiated the study of beta-plane turbulence. He predicted theinverse energy cascade into predominantly zonal modes, hence anarray of eastward--westward jets, and estimated the jet number(celebrated Rhines scale). He also proposed a $k^{-5}$ scaling lawof zonal energy spectra. Our paper reexamines scaling, spectra andzonal structure of beta-plane turbulence, based on theoreticalpredictions and numeric experiments. We show that the inversecascade gives rise to strong organized zonal jets that evolve apeculiar frontal-band (``saw-tooth'') vorticity profile. Suchstructure affects all spectral properties of the system, bycreating organized sequences of spectral peaks, and thus compoundsany putative ``scaling behavior''. The frontal-band structureappears consistently in all stochastically forced beta-plane flows,independent of dissipation and/or other details. But the resultingturbulent quasi-equilibrium is not unique, its gross parameters(jets number, mean vorticity gradient) retain memory of the initialstate and/or history.