This should really be a comment but it's way too long:
As the comments indicate, all we really have so far is a qualitative observation. It's quite interesting, but turning it into a precise question is nontrivial, and of course we have to do that before we can get an answer.
So let me take a stab at precisiating the phenomenon we see - or rather, precisiating one particular question about whether a general pattern we seem to observe actually holds. Specifically, I want to ask: "Do all the loops roll clockwise?"
There are a couple things we need to pay attention to:
We need a precise definition of "loop," and we then need a way to talk about the motion of a loop over time.
Whatever we whip up should be parameterization-invariant: we're really paying attention to the map $$\Gamma:c\mapsto\{(x,y):\exists t(x=\sum_{n=0}^\infty \frac{\cos(2^nt+cn)}{2^n}\quad\mbox{and}\quad y=\sum_{n=0}^\infty \frac{\sin(2^nt+cn)}{2^n})\},$$ or if you prefer "successive horizontal slices" of the surface $$\{(x,y,z):\exists t(x=\sum_{n=0}^\infty \frac{\cos(2^nt+zn)}{2^n}\quad\mbox{and}\quad y=\sum_{n=0}^\infty \frac{\sin(2^nt+zn)}{2^n})\}\subseteq\mathbb{R}^3.$$
So our definition will probably use the function $$F(t,c)=(\sum_{n=0}^\infty \frac{\cos(2^nt+cn)}{2^n},\sum_{n=0}^\infty \frac{\sin(2^nt+cn)}{2^n})$$ but it will ultimately "lose information."
OK, so let's start with the notion of a loop at time $c$. There are a couple things this could mean.
From algebraic topology we have the simple notion of a continuous function $S_1\rightarrow\Gamma(c)$ (or rather, an appropriate equivalence class of such functions). This notion is quite nice; for one thing, it produces a groupoid for each $c$, and so overall a "continuously varying groupoid."
However, that's a bit abstract; more importantly, it ignores details like direction of movement, which we definitely care about here. So while it's a very natural thing to look at, I'm not sure it's quite the right idea here.
Instead, I want to go a bit more concrete:
A loop at time $c$ is just a pair of numbers $a<b$ such that $F(a,c)=F(b,c)$ but for each $z\in (a,b)$ we have $F(z,c)\not=F(a,c)$. (Note that this does refer to the parameterization; that will go away later, however. The key feature here is that the parameterization $F$ is "locally injective in $c$," that is for each $t,c$ there is some $\epsilon>0$ such that $F\upharpoonright (t-\epsilon, t+\epsilon)\times\{c\}$ is injective.)
To talk about the motion of a loop over time, we'll introduce the notion of a bubble. Basically, a bubble is a "continuously varying nontrivial loop" which is maximal in the sense that if we could keep going either forward or backward in $c$ we do so. More precisely, we'll say that a prebubble consists of a tuple $(I,f,g)$ where $I$ is an open interval (which is allowed to extend infinitely in one or both directions) and $f,g$ are injective functions defined on $I$ such that $\langle f(c),g(c)\rangle$ is a loop for all $c\in I$. A prebubble doesn't necessarily "tell the whole story" of a loop's lifetime, and so we further define a bubble to be a prebubble $(I,f,g)$ such that there is no other prebubble $(\hat{I},\hat{f},\hat{g})$ with $I\subsetneq I'$ and $\hat{f}\upharpoonright I=f, \hat{g}\upharpoonright I=g$.
Now suppose I have a bubble $\beta:=(I,f,g)$. At each time $c\in I$ there is a "special point" in this bubble, $p_\beta(c)=F(f(c),c)=F(g(c),c)$. And via some polar coordinate tedium the following question can be phrased precisely:
If $\beta=(I,f,g)$ is a bubble, need $p_\beta(c)$ always be moving clockwise as $c$ increases?
(The tedium is basically that the argument of a point is many-valued. It's easily dealt with, though.)
Note that as desired, at this point everything is parameterization-free - the above works as long as we fix any parameterization which is locally injective in $c$.
Now that we have a precise question, I'm sure finding the answer will be basically trivial so I'll leave that as an exercise to the reader. :P
