Aligning neural dynamics with movements is a fundamental goal of neuroscience and brain-machine interfaces. However, we still lack a dimensionality reduction method that can align low-dimensional latent dynamics with movement. To fill this gap, we propose Neural Embedding Ranks (NER), which embed neural dynamics into a 3D latent space and contrast the embeddings based on movement ranks. Essentially, NER learns to regress continuous representations of neural dynamics (i.e., embeddings) on continuous movement.We apply NER and six other dimensionality reduction techniques to neurons in the primary motor cortex (M1), dorsal premotor cortex (PMd), and primary somatosensory cortex (S1) as monkeys perform reaching tasks. Only NER aligns latent dynamics with both hand position and direction, visualizable in 3D. NER reveals consistent latent dynamics in M1 and PMd across sixteen sessions over one year. A linear regression decoder with NER explains 86% and 97% of the variance in velocity and position, respectively. Linear models trained on data from one session can decode velocity, position, and direction in held-out test data from different dates and areas (64%, 88%, and 90%).NER also reveals distinct latent dynamics in S1 during consistent movements and in M1 when the monkey performs curved reaching tasks.Our code is uploaded.