Given a part design, the task of manufacturing process classification identifies an appropriate manufacturing process to fabricate it. Our previous research proposed a large dataset for manufacturing process classification and achieved accurate classification results based on a combination of a convolutional neural network (CNN) and the heat kernel signature (HKS) for triangle meshes. In this paper, we constructed a classification method based on rotation invariant shape descriptors and a neural network, and it achieved better accuracy than all previous methods. This method uses a point cloud part representation, in contrast to the triangle mesh representation used in our previous work. The first step extracted rotation invariant features consisting of a set of distances between points in the point cloud. Then, the extracted shape descriptors were fed into a CNN for the classification of manufacturing processes. In addition, we provide two visualization methods for interpreting the intermediate layers of the neural network. Last, the performance of the method was tested on some ambiguous examples and their performances were consistent with expectations. In this paper, we have considered only shape information, while non-shape information like materials and tolerances were ignored. Additionally, only parts that require one manufacturing process were considered in this research. Our work demonstrates that part shape attributes alone are adequate for discriminating between different manufacturing processes considered.