  This study investigates the topological behavior of a continuous elastic
phononic structure characterized by a 3-term Kekul\'e distortion. The elastic
waveguide consists of a hexagonal unit cell whose geometric dimensions are
intentionally perturbed according to a generalized Kekul\'e scheme. The
resulting structure exhibits an effective Hamiltonian that resembles a quantum
spin Hall system, hence suggesting that the waveguide can support helical
topological edge states. Using first-principles calculations, we show the
existence of nondegenerate pseudospin states and a very peculiar 6-lobe
pseudospin texture and Berry curvature pattern. Important insights are also
provided concerning the topological states in the Kekul\'e lattice, so far
considered indistinguishable, and their critical role in enabling unique
gapless edge states, typically not achievable in phononic systems.
