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Suppose that D is a planar Jordan domain and x and y are distinct boundarypoints of D. Fix \kappa \in (4,8) and let \eta\ be an SLE_\kappa process from xto y in D. We prove that the law of the time-reversal of \eta is, up toreparameterization, an SLE_\kappa process from y to x in D. More generally, weprove that SLE_\kappa(\rho_1;\rho_2) processes are reversible if and only ifboth \rho_i are at least \kappa/2-4, which is the critical threshold at orbelow which such curves are boundary filling.  Our result supplies the missing ingredient needed to show that for all \kappa\in (4,8) the so-called conformal loop ensembles CLE_\kappa\ are canonicallydefined, with almost surely continuous loops. It also provides an interestingway to couple two Gaussian free fields (with different boundary conditions) sothat their difference is piecewise constant and the boundaries between theconstant regions are SLE_\kappa curves.

Imaginary geometry III: reversibility of SLE_\kappa\ for \kappa \in (4,8)