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Robot-to-human handovers are common exercises in many robotics application domains. The requirements of handovers may vary across these different domains. In this paper, we first devised a taxonomy to organize the diverse and sometimes contradictory requirements. Among these, task-oriented handovers are not well-studied but important because the purpose of the handovers in human-robot collaboration (HRC) is not merely to pass an object from a robot to a human receiver, but to enable the receiver to use it in a subsequent tool-use task. A successful task-oriented handover should incorporate task-related information - orienting the tool such that the human can grasp it in a way that is suitable for the task. We identified multiple difficulty levels of task-oriented handovers, and implemented a system to generate handovers with novel tools on a physical robot. Unlike previous studies on task-oriented handovers, we trained the robot with tool-use demonstrations rather than handover demonstrations, since task-oriented handovers are dependent on the tool usages in the subsequent task. We demonstrated that our method can adapt to all difficulty levels of task-oriented handovers, including tasks that matched the typical usage of the tool, tasks that required an improvised or unusual usage of the tool, and tasks where the handover was adapted to the pose of a manipulandum.