I have some difficulties understanding asymptotics in general.
Is $O(n)$ the same as $O(-n)$?
Is $O(f(n))$ the same as $O(cf(n))$ even though we know that $f(n)\leq 1$ for all $n$?
I know the general Standard rules you can find e.g. on Wikipedia, but I don't have a very good intuitive understanding of these things.
The answer to both questions is yes. The second question of course requires that $c\neq 0$, since it is obvious that $O(f(n))\neq O(0)$ for any function $f$.
For example, a function $f$ is $O(n)$, iff $|f(n)| < M|n|$ for some $M$ and large enough $n$. But a function is $O(-n)$ if $|f(n)| < M|-n|$ for some $M$, which is obviously the same.
The same goes for $O(cf(n))$, as long as $c\neq 0$. If you have a function $g\in O(f(n))$, that means that $|g(n)| < M|f(n)|$ for some $M$. But that also means that $|g(n)|< \frac{M}{|c|}|cf(n)|$, so $g\in O(cf(n))$.