  Very few explicit inflationary scenarios are known to generate a large
bispectrum of orthogonal shape. Dirac-Born-Infeld Galileon inflation, in which
an induced gravity term is added to the DBI action, is one such model. We
formulate it in the language of the effective field theory of inflation by
identifying the unitary gauge operators that govern the behavior of its
cosmological fluctuations. We show how to recover rather easily from this its
power spectrum and bispectrum, which we calculated previously using standard
cosmological perturbation theory. We push our calculations up to the
determination of the fourth-order action and of the trispectrum, in which
shapes absent in k-inflation arise due to the presence of higher-order
derivative operators. We finally discuss the combined constraints set on this
model by current observational bounds on the bispectrum and trispectrum.
