Most published shear-wave (VS) velocity models of cratons include a VS increase with depth below the Moho, with a maximum at 100-150 km depth. This feature is seen in regional and global 3D tomography models and in regional 1D VS profiles. Taken at face value, it implies an oscillatory geotherm, with a ubiquitous temperature decrease below the Moho, which is implausible. The VS increase with depth has thus been attributed to strong compositional layering in the lithosphere. One recent model postulated widespread hydration and metasomatism in the uppermost cratonic mantle, decreasing VS just below the Moho. An alternative model suggested a strong enrichment of the lower cratonic lithosphere in eclogite and diamond, increasing VS but implying an unusual lithospheric composition. Here, we assemble a representative dataset of phase-velocity curves of Rayleigh and Love surface waves for cratons globally, including the all-craton averages, averages over regions in southern Africa, and interstation measurements elsewhere. We perform both thermodynamic and purely seismic inversions and show that the sub-Moho VS increase is not required by the data. Models with equilibrium, conductive lithospheric geotherms and ordinary, depleted-peridotite compositions fit the surface-wave data fully. A model-space mapping quantifies the strong trade-off between seismic velocities just below the Moho and at 100-150 km depth, which is the cause of the ambiguity. The reason why most seismic models contain a VS increase with depth below the Moho is regularization that penalizes deviations from global average reference models, which are much slower than cratonic VS profiles.