Learning 3D point sets with rotational invariance is an important and challenging problem in machine learning. Through rotational invariant architectures, 3D point cloud neural networks are relieved from requiring a canonical global pose and from exhaustive data augmentation with all possible rotations. In this work, we introduce a rotational invariant neural network by combining recently introduced vector neurons with self-attention layers to build a point cloud vector neuron transformer network (VNT-Net). Vector neurons are known for their simplicity and versatility in representing SO(3) actions and are thereby incorporated in common neural operations. Similarly, Transformer architectures have gained popularity and recently were shown successful for images by applying directly on sequences of image patches and achieving superior performance and convergence. In order to benefit from both worlds, we combine the two structures by mainly showing how to adapt the multi-headed attention layers to comply with vector neurons operations. Through this adaptation attention layers become SO(3) and the overall network becomes rotational invariant. Experiments demonstrate that our network efficiently handles 3D point cloud objects in arbitrary poses. We also show that our network achieves higher accuracy when compared to related state-of-the-art methods and requires less training due to a smaller number of hyperparameters in common classification and segmentation tasks.