We present high-resolution Very Large Telescope/Ultraviolet Visual Echelle Spectrograph observations of 56 stars in the extended horizontal-branch (EHB) of the Galactic globular clusters NGC 1904, NGC 2808, NGC 6093, and NGC 7078. Our data reveal for the first time the presence in NGC 1904 of a sizable population of fast (vsini>=20 km s-1) horizontal-branch (HB) rotators, confined to the cool end of the EHB, similar to that found in M13. We also confirm the fast rotators already observed in NGC 7078. The cooler stars (Teff<11,500 K) in these three clusters show a range of rotation rates, with a group of stars rotating at ~15 km s-1 or less and a fast rotating group at ~30 km s-1. Apparently, the fast rotators are relatively more abundant in NGC 1904 and M13 than in NGC 7078. No fast rotators have been identified in NGC 2808 and NGC 6093. All the stars hotter than Teff~11,500 K have projected rotational velocities of vsini<12 km s-1, but less than 20% have vsini<2 km s-1. The connection between photometric gaps in the HB and the change in the projected rotational velocities is not confirmed by the new data. However, our data are consistent with a relation between this discontinuity and the HB jump. We discuss a number of possibilities for the origin of the stellar rotation distribution along the HB. We conclude that none of them can yet provide a satisfactory explanation of the observations. Based on observations with the ESO Very Large Telescope Ultraviolet Visual Echelle Spectrograph at the Paranal Observatory, Chile.