The genetic basis of fluctuating asymmetry (FA), a measure of random deviations from perfect bilateral symmetry, has been the subject of much recent work. In this paper we compare two perspectives on the quantitative genetic analysis of FA and directional asymmetry (DA). We call these two approaches the character-state model and the environmental responsiveness model. In the former approach, the right and left sides are viewed as separate traits whose genetic coupling is manifested by the genetic correlation. This model leads to the relationship, h²_DA = h²[(1 − r_A)/(1 − r_P)], where h² is the heritability of each component trait (assumed to be the same), r_A and r_P are the genetic and phenotypic correlations between traits, respectively. Simulation shows that, under this model, the heritability of FA is considerably less than that of DA, except when heritabilities are very close to zero. The environmental responsiveness model permits genetic variance in FA even when the genetic correlation between traits is 1. Simulation shows that under this model the heritability of FA can be uncoupled from that of DA. The additive and nonadditive components of the component (right and left) traits, their DA and FA values are estimated using a diallel cross of seven inbred lines of the sand cricket, Gryllus firmus. Four leg measurements were made and both the individual DA and FA values and the compound measures DASUM and CFA estimated. The heritabilities of the compound measures are slightly larger than the individual estimates. Dominance variance is observed in the individual traits but predicted to be an even smaller component of the phenotypic variance than the additive genetic variance. The estimated values confirm this, although a previous study has demonstrated that dominance variance is present. Because the heritabilities of FA are generally larger than those of DA, which never exceed 0.02, the environmental responsiveness model is more consistent with the data than the character-state model. A review of other data suggests that both sources of variation might be found in some species.