Developmental processes can be viewed as complex dynamical systems, where a small set of simple rules of cellular and physico-chemical interaction can interact to generate a complex morphology. We analyze how the structure of these mechanistic rules of development can constrain processes of morphological evolution. In particular, we focus on the issue of mathematical bifurcations in dynamical systems and how the presence of bifurcations implies discontinuity and directionality in phyletic transformations. We illustrate these issues by presenting a model of early epithelial morphogenesis that shows how a wide variety of seemingly unrelated processes (e.g., feathers, scales, glands, and hair follicles) can be interpreted as variations along a common underlying developmental program. We conclude that consideration of epigenetic interactions is essential to understand the origin of morphological novelties.