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The growing demand for virtual reality (VR) applications requires wireless systems to provide a high transmission rate to support 360-degree video streaming to multiple users simultaneously. In this paper, we propose an intelligent reflecting surface (IRS)-aided rate-splitting (RS) VR streaming system. In the proposed system, RS facilitates the exploitation of the shared interests of the users in VR streaming, and IRS creates additional propagation channels to support the transmission of high-resolution 360-degree videos. IRS also enhances the capability to mitigate the performance bottleneck caused by the requirement that all RS users have to be able to decode the common message. We formulate an optimization problem for maximization of the achievable bitrate of the 360-degree video subject to the quality-of-service (QoS) constraints of the users. We propose a deep deterministic policy gradient with imitation learning (Deep-GRAIL) algorithm, in which we leverage deep reinforcement learning (DRL) and the hidden convexity of the formulated problem to optimize the IRS phase shifts, RS parameters, beamforming vectors, and bitrate selection of the 360-degree video tiles. We also propose RavNet, which is a deep neural network customized for the policy learning in our Deep-GRAIL algorithm. Performance evaluation based on a real-world VR streaming dataset shows that the proposed IRS-aided RS VR streaming system outperforms several baseline schemes in terms of system sum-rate, achievable bitrate of the 360-degree videos, and online execution runtime. Our results also reveal the respective performance gains obtained from RS and IRS for improving the QoS in multiuser VR streaming systems.