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Developments in phonological theory have led to the recognition that phonological representations have a layered constituent structure. Many, perhaps all, of these constituents contain elements which can be identified as heads. Heads enter into various kinds of relations with their dependents. In this article, we identify a phenomenon which is quite pervasive in every part of phonology which has heads and dependents, namely, the existence of head–dependent asymmetries (henceforth HDAs). While various particular manifestations of these asymmetries are well known and have been much studied, this is the first attempt, to our knowledge, to unite a broad range of seemingly different phenomena under one heading. We identify various types of HDAs, and propose constraints on possible HDAs.Most importantly, we distinguish between HDAs that involve complexity, and those that involve visibility. These have properties which potentially contradict each other. We propose that they apply in fundamentally different types of cases: unlike complexity HDAs, visibility HDAs are limited to mappings from one phonological plane to another, and so are related to the notion of projection (cf. Vergnaud 1977).We also wish to show that an understanding of HDAs reveals general structural principles that play a role in diverse phenomena at various levels of the phonological hierarchy. For example, the fact noted in the Optimality Theory literature that certain positions tend to be more ‘faithful’ to underlying specifications (Beckman 1998) is a consequence of the fact that heads allow more complexity. These principles act as constraints on possible constraints, and on possible mappings from one plane to another.