  Following recent work, we discuss waves in a warm ideal two-fluid plasma
consisting of electrons and ions starting from a completely general, ideal
two-fluid dispersion relation. The plasma is characterised by five variables:
the electron and ion magnetisations, the squared electron and ion sound speeds,
and a parameter describing the angle between the propagation vector and the
magnetic field. The dispersion relation describes 6 pairs of waves which we
label S, A, F, M, O, and X. Varying the angle, it is argued that parallel and
perpendicular propagation (with respect to the magnetic field) exhibit unique
behaviour. This behaviour is characterised by the crossing of wave modes which
is prohibited at oblique angles. We identify up to 6 different parameter
regimes where a varying number of exact mode crossings in the special parallel
or perpendicular orientations can occur. We point out how any ion-electron
plasma has a critical magnetisation (or electron cyclotron frequency) at which
the cutoff ordering changes, leading to different crossing behaviour. These are
relevant for exotic plasma conditions found in pulsar and magnetar
environments. Our discussion is fully consistent with ideal relativistic MHD
and contains light waves. Additionally, exploiting the general nature of the
dispersion relation, phase and group speed diagrams can be computed at
arbitrary wavelengths for any parameter regime. Finally, we recover earlier
approximate dispersion relations that focus on low-frequency limits and make
direct correspondences with some selected kinetic theory results.
