sum of all positive integers less than $n$ and relatively prime to $n$
It’s true for all $n>2$. The reason is that if $k\in\{1,\ldots,n-1\}$ is relatively prime to $n$, so is $n-k$, so the integers that you’re adding can be combined into $\frac{\varphi(n)}2$ pairs whose members sum to $n$. If $n>2$, $\frac{n}2$ is never relatively prime to $n$, so you really do get pairs $\{k,n-k\}$.
See OEIS A023896 for some references.