  We present high-resolution measurements of the relative length change as a
function of temperature of the organic charge-transfer salt
$\kappa$-(BEDT-TTF)$_2$Hg(SCN)$_2$Cl. We identify anomalous features at $T_g
\approx\,63$ K which can be assigned to a kinetic glass-like ordering
transition. By determining the activation energy $E_A$, this glass-like
transition can be related to conformational degrees of freedom of the ethylene
endgroups of the organic building block BEDT-TTF. As opposed to other
$\kappa$-(BEDT-TTF)$_2X$ salts, we identify a peculiar ethylene endgroup
ordering in the present material in which only one of the two
crystallographically inequivalent ethylene endgroups is subject to glass-like
ordering. This experimental finding is fully consistent with our predictions
from $ab\,initio$ calculations from which we estimate the energy differences
$\Delta E$ and the activation energies $E_A$ between different conformations.
The present results indicate that the specific interaction between the ethylene
endgroups and the nearby anion layers leads to different energetics of the
inequivalent ethylene endgroups, as evidenced by different ratios $E_A/\Delta
E$. We infer that the ratio $E_A/\Delta E$ is a suitable parameter to identify
the tendency of ethylene endgroups towards glass-like freezing.
