  We discuss new consistency relations for single field models of inflation
capable of generating primordial black holes (PBH), and their observational
implications for CMB $\mu$-space distortions. These inflationary models include
a short period of non-attractor evolution: the scale-dependent profile of
curvature perturbation is characterized by a pronounced dip, followed by a
rapid growth leading to a peak responsible for PBH formation. We investigate
the squeezed and the collapsed limits of three and four point functions of
curvature perturbation around the dip, showing that they satisfy consistency
relations connecting their values to the total amplification of the curvature
spectrum, and to the duration of the non-attractor era. Moreover, the
corresponding non-Gaussian parameters are scale-dependent in proximity of the
dip, with features that again depend on the amplification of the spectrum. For
typical PBH scenarios requiring an order ${\cal O}(10^7)$ enhancement of the
spectrum from large towards small scales, we generally find values $f_{\rm
NL}^{\rm sq}\,=\,{\cal O}(10)$ and $\tau_{\rm NL}^{\rm col}\,=\,{\cal O}(10^3)$
in a range of scales that can be probed by CMB $\mu$-space distortions. Using
these consistency relations, we carefully analyze how the scale-dependence of
non-Gaussian parameters leads to characteristic features in $\langle \mu T
\rangle$ and $\langle \mu \mu \rangle$ correlators, providing distinctive
probes of inflationary PBH scenarios that can be tested using well-understood
CMB physics.
