An inverse cascade, energy transfer to progressively larger scales, is a salient feature of two-dimensional turbulence. If the cascade reaches the system scale, it creates a coherent flow expected to have the largest available scale and conform with the symmetries of the domain. In a square doubly periodic domain, the mean flow is expected to take the form of a vortex dipole. The velocity profile of a corresponding single vortex was recently obtained analytically and subsequently confirmed numerically. I will describe the next step in the derivation: using the mean velocity profile to predict features of the turbulent fluctuations.  I will also address the mean flow in a doubly periodic (non-square) rectangle. For a rectangle, the mean flow with zero total momentum was believed to be unidirectional, with two jets along the short side. I will describe how direct numerical simulations reveal that neither the box symmetry is respected nor the largest scale is realized: the flow is never purely unidirectional since the inverse cascade produces coherent vortices, whose number and relative motion are determined by the aspect ratio. This spontaneous symmetry breaking is closely related to the hierarchy of averaging times. Long-time averaging restores translational invariance due to vortex wandering along one direction, and gives jets whose profile, however, can be deduced neither from the largest-available-scale argument, nor from the often employed maximum-entropy principle or quasi-linear approximation.