One of the central questions in motor neuroscience is how the population of motor cortical neurons control voluntary movement. Much of our current understanding of this question is based on so-called center-out reaching tasks that involve reaching from a single start location to various spatial targets --- a situation that confounds reach direction with limb posture at the spatial target. Because of this confound, previous studies lack the required condition variability to probe the geometry of neural dynamics (NDs) for movements across different postures and extents. To address this confound, we trained two macaques to move their arm between all possible combinations of five targets located on the vertices and at the center of a rectangle. High-density recordings from the primary (M1) and pre-motor (PMd) cortex revealed an elegant compositional arrangement of NDs for movement with different postures, extents, and directions with two key features: (1) a posture subspace with attractor points for each spatial target. These attractors were visited whenever the arm rested in its respective target before or after the reach; (2) rotational dynamics that linked the attractor points. These rotational patterns were aligned such that more similar rotational dynamics were associated with more similar reach directions. To gain mechanistic insight into the nature of this compositional arrangement, we trained recurrent neural networks (RNNs) to control a realistic biomechanical model arm across a 2D workspace. We then probed the RNN’s NDs on the same 5-target point-to-point task. We found a striking similarity between the brain and RNN with both key compositional elements observed in RNNs under a wide range of network parameters. Our work offers a broader understanding of how the geometry of NDs in the primate motor cortex is related to movement parameters like posture, extent, and direction, with important implications for future modeling and brain-computer interface design.