The classical twin design (CTD) is the most common method used to infer genetic and environmental causes of phenotypic variation. As has long been acknowledged, different combinations of the common environment/assortative mating, and additive, dominant, and epistatic genetic effects can lead to the same observed covariation between twin pairs, meaning that there is an inherent indeterminacy in parameter estimates arising from the CTD. The CTD circumvents this indeterminacy by assuming that higher-order epistasis is negligible and that the effects of either dominant genetic variation or the common environment are nonexistent. These assumptions, however, lead to consistent biases in parameter estimation. The current paper quantifies these biases and discusses alternative strategies for dealing with parameter indeterminacy in twin designs. One strategy is to model the similarity among other relatives in addition to twins (extended twin-family designs), which reduces but does not eliminate indeterminacy in parameter estimates. A more general strategy, applicable to all twin designs, is to present the parameter indeterminacy explicitly, as in a graph. Presenting the space of mathematically equally likely parameter values is important, not only because it aids the proper interpretation of twin design findings, but also because it keeps behavioral geneticists themselves mindful of methodological assumptions that can easily go unexamined.