I was looking for the same thing, without much success.
The best paper I found is "ON VANISHING SUMS OF ROOTS OF UNITY" by Lam and Leung. They show that if n has at most two prime divisors then a vanishing sum of n-th roots of unity indeed decomposes into the "obvious ones". In the paper, you can also find a counterexample for n having three prime divisors.
Even with many divisors, it is possible that a vanishing sum decomposes into "less obvious ones", but this doesn't seem to be known.