Fetal echocardiography is essential for detecting congenital heart disease (CHD), facilitating pregnancy management, optimized delivery planning, and timely postnatal interventions. Among standard imaging planes, the four-chamber (4CH) view provides comprehensive information for CHD diagnosis, where clinicians carefully inspect the end-diastolic (ED) and end-systolic (ES) phases to evaluate cardiac structure and motion. Automated detection of these cardiac phases is thus a critical component toward fully automated CHD analysis. Yet, in the absence of fetal electrocardiography (ECG), manual identification of ED and ES frames remains a labor-intensive bottleneck. We present ORBIT (Orientation-Robust Beat Inference from Trajectories), a self-supervised framework that identifies cardiac phases without manual annotations under various fetal heart orientation. ORBIT employs registration as self-supervision task and learns a latent motion trajectory of cardiac deformation, whose turning points capture transitions between cardiac relaxation and contraction, enabling accurate and orientation-robust localization of ED and ES frames across diverse fetal positions. Trained exclusively on normal fetal echocardiography videos, ORBIT achieves consistent performance on both normal (MAE = 1.9 frames for ED and 1.6 for ES) and CHD cases (MAE = 2.4 frames for ED and 2.1 for ES), outperforming existing annotation-free approaches constrained by fixed orientation assumptions. These results highlight the potential of ORBIT to facilitate robust cardiac phase detection directly from 4CH fetal echocardiography.