  We compute the rotational anisotropy of the free energy of $\alpha$-RuCl$_3$
in an external magnetic field. This quantity, known as the magnetotropic
susceptibility, $k$, relates to the second derivative of the free energy with
respect to the angle of rotation. We have used approximation-free,
auxiliary-field quantum Monte Carlo simulations for a realistic model of
$\alpha$-RuCl$_3$ and optimized the path integral to alleviate the negative
sign problem. This allows us to reach temperatures down to $30~\rm{K}$ -- an
energy scale below the dominant Kitaev coupling. We demonstrate that the
magnetotropic susceptibility in this model of $\alpha$-RuCl$_3$ displays unique
scaling, $k = Tf(B/T)$, with distinct scaling functions $f$ at high and low
temperatures. In comparison, for the XXZ Heisenberg model, the scaling $k =
Tf(B/T)$ breaks down at a temperature scale where the uniform spin
susceptibility deviates from the Curie law (i.e. at the energy scale of the
exchange interactions) and never recovers at low temperatures. Our findings
suggest that correlations in $\alpha$-RuCl$_3$ lead to degrees of freedom that
respond isotropically to a magnetic field. One possible interpretation for the
apparent scale-invariance observed in experiments could be fractionalization of
the spin degrees of freedom in the extended Kitaev model.
