  In single-field inflationary models with a low sound speed, the orthogonal
shape of the primordial bispectrum arises due to partial cancellations between
equilateral-type shapes. This fact allows for a speed of sound c_s as low as
about 0.01, which is actually weakly preferred by WMAP data. For such values,
the trispectrum, scaling like 1/c_s^4, is of order 10^8 and is therefore
comparable to, and greater than, the 1 sigma observational bound t_NL^eq=(-3.11
+- 7.5)*10^6. Hence, the trispectrum is already constraining inflationary
mechanisms candidates for generating an orthogonal bispectrum at the level
hinted in WMAP data. If this signal persists in imminent Planck data, most of
the parameter space of the simplest effective field theory of inflation will be
under observational pressure, while a dedicated analysis will be needed for the
substantial fraction of parameter space where we show that a qualitatively new,
orthogonal, trispectrum naturally arises.
