The bivariate copulas that describe the dependencies and partial dependencies of lagged variables in strictly stationary, first-order GARCH-type processes are investigated. It is shown that the copulas of symmetric GARCH processes are jointly symmetric but non-exchangeable, while the copulas of processes with symmetric innovation distributions and asymmetric leverage effects have weaker h-symmetry; copulas with asymmetric innovation distributions have neither form of symmetry. Since the actual copulas are typically inaccessible, due to the unknown functional forms of the marginal distributions of GARCH processes, methods of mimicking them are proposed. These rely on constructions that combine standard bivariate copulas for positive dependence with two uniformity-preserving transformations known as v-transforms. A variety of new copulas are introduced and the ones providing the best fit to simulated data from GARCH-type processes are identified. A method of constructing tractable simplified d-vines using linear v-transforms is described and shown to coincide with the vt-d-vine model when the two v-transforms are identical.