There must be an easier proof but here is a nice approach which can indeed lead to deeper results (feel free to edit for math display, I tried):
Techniques with characteristic polynomials and roots of unity can be very powerful. I like the way that the appropriate lemmas are explained in my paper with Ethan Coven "Tiling the Integers with Translates of One Finite Set" http://arxiv.org/abs/math/9802122 or Journal of Algebra v 212 (1988) p 161-174. One does not need their full generality for this problem but perhaps for  deeper results. 

I'll sketch this result which implies what was asked for: Suppose that A and B are sets of size #A and #B so that A+B is a complete set of residues mod N=#A#B. Let p be a prime dividing N. Then exactly one of the sets has its members equally distributed mod p. 

digression: Lemma 3.2 from the paper above (not needed here) shows that at least one of the following is true:

1) No member of A-A is relatively prime to #B

2) No member of B-B is relatively prime to #A 
end of digression

Consider the corresponding polynomials $A(x)=\sum_{a \in A}x^a$ and $B(x)=\sum_{b \in B}x^b$. Then 

i) A(1)=#A and B(1)=#B

ii) A(x)B(x) is a sum of N distinct powers of x, one from each residue class.

iii) $A(x)B(x)=(x^N-1)Q(x)+\frac{x^N-1}{x-1}$ for some polynomial Q(x).

iv) Every irreducible polynomial dividing $\frac{x^N-1}{x-1}$  divides at least one of $A(x)$ and $B(x)$
 

As an example consider A={0,9,13,16,29,32} B={0,10,12,22,24,34} with A+B a complete set of residues mod N=36.

  $$\frac{x^36-1}{x-1}=(x+1)(x^2+x+1)(x^2+1)(x^2-x+1)(x^4+x^2+1)(x^4-x^2+1)(x^18-x^9+1)$$

 evaluated at $x=1$ this becomes 36=2 * 3 * 2 * 1 * 3 * 1

In general the irreducible polynomial divisors of $\frac{x^N-1}{x-1}$ are the cyclotomic polynomials corresponding to the divisors of N. Evaluated at x=1 each is either 1 (composite divisor) or a prime p (prime power divisor) and the primes have product N. Since A(1)B(1)=N and A(x)B(x) is divisible by all the prime power cyclotomic divisors of $\frac{x^N-1}{x-1}$ and these evaluated at 1 also have product N, each divides just one of A(x) or B(x) and all other polynomial divisors evaluate to 1 at 1. In particular: for each prime divisor of N, only one of A(X), B(x) divides by $\frac{x^p-1}{x-1}$ and only that one has corresponding set equidistributed mod p.

In our example A is a complete set of residues mod 6 so A(x) divides by (1+x) and by (1+x+x^2). 
Since A(1)=6 , A(x) can't  have either of (1+x^2) and (1+x^2+x^4) as factors. But they do divide A(x)B(x) and hence they divide B(x). This means that neither  (1+x) nor (1+x+x^2) can divide B(x), again since B(1)=6. Hence, B is **not** equidistributed mod 2 (or mod 3) and certainly not mod 6.

By the way, $B(x)=(X^10+1)(x^24+x^12+1)$ and $A(x)=(x^13+1)(x^32+x^16+1)$ (mod $x^36-1$)