  Let $S$ be the random walk obtained from "coin turning" with some sequence
$\{p_n\}_{n\ge 1}$, as introduced in [6]. In this paper we investigate the
scaling limits of $S$ in the spirit of the classical Donsker invariance
principle, both for the heating and for the cooling dynamics.
  We prove that an invariance principle, albeit with a non-classical scaling,
holds for "not too small" sequences, the order const$\cdot n^{-1}$ (critical
cooling regime) being the threshold. At and below this critical order, the
scaling behavior is dramatically different from the one above it. The same
order is also the critical one for the Weak Law of Large Numbers to hold.
  In the critical cooling regime, an interesting process emerges: it is a
continuous, piecewise linear, recurrent process, for which the one-dimensional
marginals are Beta-distributed.
  We also investigate the recurrence of the walk and its scaling limit, as well
as the ergodicity and mixing of the $n$th step of the walk.
