A famous theorem by Sklar (1959) provides an elegant and useful way to look at multivariate dependence structures. This paper explores the construction of copulas from nonmonotone transformations applied to the components of random vectors whose marginals are uniform on ($0,1$). This approach allows the creation of new families of multivariate copulas that generalize the chi-square, Fisher, squared and V-copulas, to name a few. The properties of the resulting dependence structures are studied, including tail dependence and tail asymmetry. The usefulness of the models created is illustrated for standard multivariate dependence modeling, nonmonotone copula regression and spatial dependence.