The full title for this question should be On the many patterns of Euler totient function manipulations and their tendency towards symmetry at primorials, (but didn't want to take up too much room). For the rest of the question, $\#_n$ will represent the $n$th primorial.
Question / observations
A plot of $n/\phi(n)$ for $1\leq n\leq \#_4$ groups each $n$ that has an identical $\phi(n)$ value:
It can be seen by drawing reflected lines in $\#_4/2$ there is a tendency towards a symmetry. For the cases where $|n/\phi(n)-(\#_4-n)/\phi(\#_4-n)|=0$ where $n$ is prime are clearly the additive Goldbach prime partitions of $\#_4.$
The tendency for all $|n/\phi(n)-(\#_k-n)/\phi(\#_k-n)|\rightarrow 0$ for some $k$ shows a tendency towards symmetry:
At the primorials, for prime values only of $|n/\phi(n)-(\#_k-n)/\phi(\#_k-n)|$ looks like this for $k=5,$ surprisingly regular in comparison to non-primmorial multiples (right):
Similarly, limiting $n$ to values of all equal $\phi(N)$ where $N$ is a primorial results in clear patterns (left), whereas for non-primorial multiples, a completely diferent behaviour is exhibited (right):
Presumably, many of these patterns have simple explanations, but is something deeper going on here? I can't explain many of the patterns occurring here - in particular, the striking differences between primorials and nion-primorials. Can someone help to explain what is going on in some of these images?