Out-of-plane tissue deformations are key morphogenetic events during plant and animal development that generate 3D shapes, such as flowers or limbs. However, the mechanisms by which spatiotemporal patterns of gene expression modify cellular behaviours to generate such deformations remain to be established. We use the Snapdragon flower as a model system to address this problem. Combining cellular analysis with tissue-level modelling, we show that an orthogonal pattern of growth orientations plays a key role in generating out-of-plane deformations. This growth pattern is most likely oriented by a polarity field, highlighted by PIN1 protein localisation, and is modulated by dorsoventral gene activity. The orthogonal growth pattern interacts with other patterns of differential growth to create tissue conflicts that shape the flower. Similar shape changes can be generated by contraction as well as growth, suggesting tissue conflict resolution provides a flexible morphogenetic mechanism for generating shape diversity in plants and animals.