Flipping and cyclic shifting of binary aggregation functions

We introduce two types of transformations of random variables, called flipping and cyclic shifting. As these transformations preserve monotonicity at the level of univariate cumulative distribution functions, they can be used to develop corresponding coordinate-wise transformations of binary aggregation functions. We lay bare the admissibility of these transformations, i.e. the necessary and sufficient conditions under which they result in a binary aggregation function. We investigate which additional properties, such as the 1-Lipschitz property and 2-increasingness, entail these admissibility conditions. Moreover, we point out which of these properties are preserved under flipping and/or cyclic shifting. Interestingly, quasi-copulas remain quasi-copulas under flipping, while copulas remain copulas under flipping as well as under cyclic shifting.