Compliant and soft hands have gained a lot of attention in the past decade because of their ability to adapt to the shape of the objects, increasing their effectiveness for grasping. However, when it comes to grasping highly flexible objects such as textiles, we face the dual problem: it is the object that will adapt to the shape of the hand or gripper. In this context, the classic grasp analysis or grasping taxonomies are not suitable for describing textile objects grasps. This work proposes a novel definition of textile object grasps that abstracts from the robotic embodiment or hand shape and recovers concepts from the early neuroscience literature on hand prehension skills. This framework enables us to identify what grasps have been used in literature until now to perform robotic cloth manipulation, and allows for a precise definition of all the tasks that have been tackled in terms of manipulation primitives based on regrasps. In addition, we also review what grippers have been used. Our analysis shows how the vast majority of cloth manipulations have relied only on one type of grasp, and at the same time we identify several tasks that need more variety of grasp types to be executed successfully. Our framework is generic, provides a classification of cloth manipulation primitives and can inspire gripper design and benchmark construction for cloth manipulation.