Most animals show external bilateral symmetry, which hinders the observation of multiple internal left–right (L/R) asymmetries that are fundamental to organ packaging and function1,2. In vertebrates, left identity is mediated by the left-specific Nodal–Pitx2 axis that is repressed on the right-hand side by the epithelial–mesenchymal transition (EMT) inducer Snail1 (refs3,4). Despite some existing evidence3,5, it remains unclear whether an equivalent instructive pathway provides right-hand-specific information to the embryo. Here we show that, in zebrafish, BMP mediates the L/R asymmetric activation of another EMT inducer, Prrx1a, in the lateral plate mesoderm with higher levels on the right. Prrx1a drives L/R differential cell movements towards the midline, leading to a leftward displacement of the cardiac posterior pole through an actomyosin-dependent mechanism. Downregulation of Prrx1a prevents heart looping and leads to mesocardia. Two parallel and mutually repressed pathways, respectively driven by Nodal and BMP on the left and right lateral plate mesoderm, converge on the asymmetric activation of the transcription factors Pitx2 and Prrx1, which integrate left and right information to govern heart morphogenesis. This mechanism is conserved in the chicken embryo, and in the mouse SNAIL1 acts in a similar manner to Prrx1a in zebrafish and PRRX1 in the chick. Thus, a differential L/R EMT produces asymmetric cell movements and forces, more prominent from the right, that drive heart laterality in vertebrates.