It is shown that for every positive integer n 2 and for every ρ > 0, there exist distribution functions F on the real line R such that: (i) F has absolute moments of all orders up to and including ρ, or up to and not including ρ, as we choose, and (ii) the convolutions F * r of F with itself are asymmetric (about the origin) for 1 r n - 1, while F * n is asymmetric. This relates to a question raised in Staudte and Tata (Proc. Amer. Math. Soc. 25 (1970) 668-670) and partially answered in Ramachandran (Sankhya A, 58 (1996) 1-7).