  We study the $TT\mu$ bispectrum, generated by correlations between Cosmic
Microwave Background temperature (T) anisotropies and chemical potential
($\mu$) distortions, and we analyze its dependence on primordial local
trispectrum parameters $g_{\rm NL}$ and $\tau_{\rm NL}$. We cross-check our
results by comparing the full bispectrum calculation with the expectations from
a general physical argument, based on predicting the shape of $\mu$-T
correlations from the couplings between short and long perturbation modes
induced by primordial non-Gaussianity. We show that $both$ $g_{\rm NL}$ and
$\tau_{\rm NL}$-parts of the primordial trispectrum source a non-vanishing
$TT\mu$ signal, contrary to the $\mu\mu$ auto-correlation function, which is
sensitive only to the $\tau_{\rm NL}$-component. A simple Fisher matrix-based
forecast shows that a futuristic, cosmic-variance dominated experiment could in
principle detect $g_{\rm NL} \sim 0.4$ and $\tau_{\rm NL} \sim 40$ using
$TT\mu$.
