Aggregation processes appear naturally in many fields of application. The formalization of such processes has been a core topic for researchers in the fuzzy set community for decades, mostly focusing on the aggregation of elements of a bounded poset (typically a bounded real interval). Recent work by the present authors has aimed at further generalizing aggregation theory so it can accommodate aggregation processes on more general structures such as multivariate data, ranking data and string data. In this work, the aggregation of circular data is explored and, in particular, two prominent examples of aggregation functions for circular data (namely the circular mean and the circular median) are presented within this revisited aggregation theory framework.