We have studied a sample of Large Magellanic Cloud red giant binaries that lie on sequence E in the period–luminosity plane. We show that their combined light and velocity curves unambiguously demonstrate that they are binaries showing ellipsoidal variability. By comparing the phased light and velocity curves of both sequence D and E variables, we show that the sequence D variation – the long secondary period – is not caused by ellipsoidal variability. We also demonstrate several further differences between stars on sequences D and E. These include differences in velocity amplitude, in the distribution of eccentricity and in the correlations of velocity amplitude with luminosity and period. We also show that the sequence E stars, unlike stars on sequence D, do not show any evidence of a mid-infrared excess that would indicate circumstellar dust.