In this article, we present a new mathematical strategy for creating flexible asymmetric continuous distributions. It is designed to introduce asymmetry into any distribution with the entire real line as support, thanks to the tuning of two parameters and an intermediate function. A wide range of intermediate functions of different types can be chosen, including a high degree of adaptability. To illustrate this strategy, we present four types of asymmetric normal distributions and four types of asymmetric Cauchy distributions. Some of them have rare properties, such as multimodality (bimodality, trimodality, and more) and abrupt angular shape for the corresponding probability density functions. These features are supported by an extensive graphical analysis. Finally, we discuss the adaptation of the strategy for distributions with different support.