  The effective field theory (EFT) of inflation has become the preferred method
for computing cosmological correlation functions of the curvature fluctuation,
$\zeta$. It makes explicit use of the soft breaking of time diffeomorphisms by
the inflationary background to organize the operators expansion in the action
of the Goldstone mode $\pi$ associated with this breaking. Despite its
ascendancy, there is another method for calculating $\zeta$ correlators,
involving the direct calculation of the so-called Horndeski action order by
order in powers of $\zeta$ and its derivatives. The question we address in this
work is whether or not the $\zeta$ correlators calculated in these seemingly
different ways are in fact the same. The answer is that the actions to cubic
order in either set of variables do indeed give rise to the same $\zeta$
bispectra, but that to make this equivalence manifest requires a careful
understanding of the non-linear transformations relating $\pi$ to $\zeta$ and
how boundary terms in the actions are affected by imposing this relation. As a
by product of our study we find that the calculations in the $\pi$ language can
be simplified considerably in a way that allows us to use only the linear part
of the $\pi-\zeta$ relation simply by changing the coefficients of some of the
operators in the EFT. We also note that a proper accounting of the boundary
terms will be of the greatest importance when computing the bispectrum for more
general initial states than the Bunch-Davies one.
