  Under the presence of anisotropic sources in the inflationary era, the
trispectrum of the primordial curvature perturbation has a very specific
angular dependence between each wavevector that is distinguishable from the one
encountered when only scalar fields are present, characterized by an angular
dependence described by Legendre polynomials. We examine the imprints left by
curvature trispectra on the $TT\mu$ bispectrum, generated by the correlation
between temperature anisotropies (T) and chemical potential spectral
distortions ($\mu$) of the Cosmic Microwave Background (CMB). Due to the
angular dependence of the primordial signal, the corresponding $TT\mu$
bispectrum strongly differs in shape from $TT\mu$ sourced by the usual $g_{\rm
NL}$ or $\tau_{\rm NL}$ local trispectra, enabling us to obtain an unbiased
estimation. From a Fisher matrix analysis, we find that, in a
cosmic-variance-limited (CVL) survey of $TT\mu$, a minimum detectable value of
the quadrupolar Legendre coefficient is $d_2 \sim 0.01$, which is 4 orders of
magnitude better than the best value attainable from the $TTTT$ CMB
trispectrum. In the case of an anisotropic inflationary model with a
$f(\phi)F^2$ interaction (coupling the inflaton field $\phi$ with a vector
kinetic term $F^2$), the size of the curvature trispectrum is related to that
of quadrupolar power spectrum asymmetry, $g_*$. In this case, a CVL measurement
of $TT\mu$ makes it possible to measure $g_*$ down to $10^{-3}$.
