The paradigm-changing possibility of collective neutrino-antineutrino oscillations was recently advanced in analogy to collective flavor oscillations. However, the amplitude for the backward scattering process ${\nu }_{p_1}{\overline{\nu }}_{p_2}\to {\nu }_{p_2}{\overline{\nu }}_{p_1}$ is helicity suppressed and vanishes for massless neutrinos, implying that there is no off-diagonal refractive index between $\nu$ and $\overline{\nu }$ of a single flavor of massless neutrinos. For a nonvanishing mass, collective helicity oscillations are possible, representing de facto $\nu -\overline{\nu }$ oscillations in the Majorana case. However, such phenomena are suppressed by the smallness of neutrino masses as discussed in the previous literature.