In the present exploratory study we investigate whether cognitive flexibility is a unitary mechanism underlying flexible behaviours across many domains or a domain-specific capacity. The literature on cognitive flexibility is divided into several research lines that do not converge. The most prominent one considers flexibility an executive function that represents the ability to switch among rules or tasks. In other research traditions it is associated with distinct components, such as the capacity to place an item into many categories (in creativity tests) or a characteristic of different cognitive or perceptual processes (e.g., flexible language use, flexibility in mathematics, perceptual flexibility). To determine whether flexibility in different domains relies on a general shared mechanism, 221 subjects from two countries (The United States and Romania, mean age 19.52 years) were tested online with several measurements from four different domains of investigation: language, mathematics, perception, and executive functions (specifically, set shifting). All tasks required some form of cognitive flexibility. In addition, we measured math anxiety to see how this relates to mathematical flexibility. The results show very few and small significant partial correlations among the tasks. They also highlight that there is no unitary overarching "executive" factor. The most prominent common factor was speed of processing for mathematical and language response times. Shifting does not seem to be a mechanism that underlies flexibility in all the investigated domains. While we acknowledge the need for replication of this study, the data suggest that the construct of shifting does not exhaust the notion of flexibility as it arises across cognitive domains.