We present a complete analysis of all wave modes in a cold pair plasma, significantly extending standard textbook treatments. Instead of identifying the maximal number of two propagating waves at fixed frequency
, we introduce a unique labelling of all 5 mode pairs described by the general dispersion relation
, starting from their natural ordering at small wavenumber
. There, the 5 pairs start off as Alfvén (A), fast magnetosonic (F), modified electrostatic (M) and electromagnetic O and X branches, and each
branch smoothly connects to large wavenumber resonances or limits. For cold pair plasmas, these 5 branches show avoided crossings, which become true crossings at exactly parallel or perpendicular orientation. Only for those orientations, we find a changed connectivity between small and large wavenumber behaviour. Analysing phase and group diagrams for all 5 wave modes, distinctly different from the Clemmow–Mullaly–Allis representation, reveals the true anisotropy of the A, M and O branches.