This paper explores the unusual hierarchy of degeneracies in the linear theory of thermoelasticity. In classical elastic wave theory all degeneracies take the form of acoustic axes, that is directions in which two or all three plane bulk waves have equal speeds. In dynamical thermoelasticity four plane harmonic waves can travel in an arbitrary direction, and there are two types of degeneracy. The first type arises when two or more waves have equal slownesses, normally complex, and the second type when the coefficient matrix of the governing system of differential equations has a repeated zero eigenvalue. Each type of degeneracy is of two possible kinds, so the number of cases in which at least one degeneracy occurs is eight. It is shown that only three of these possibilities can actually exist and in only one of them are both types of degeneracy present. The effects of thermomechanical interaction on the modes of wave propagation are then minimal. An analysis of the degeneracies, their interconnexion and their influence on the nature of thermoelastic waves occupies the first part of the paper. In the second part the relationship of classical elastodynamics to linear thermoelasticity is studied, in respect of degeneracy, by considering small thermoelastic perturbations of an acoustic axis. The underlying degeneracy is either removed by the perturbation or divided into one or two pairs of thermoelastic degeneracies. The directions in which the new degeneracies appear are determined and the properties of the associated degenerate waves discussed in detail.