Content creators often use music to enhance their videos, from soundtracks in movies to background music in video blogs and social media content. However, identifying the best music for a video can be a difficult and time-consuming task. To address this challenge, we propose a novel framework for automatically retrieving a matching music clip for a given video, and vice versa. Our approach leverages annotated music labels, as well as the inherent artistic correspondence between visual and music elements. Distinct from previous cross-modal music retrieval works, our method combines both self-supervised and supervised training objectives. We use self-supervised and label-supervised contrastive learning to train a joint embedding space between music and video. We show the effectiveness of our approach by using music genre labels for the supervised training component, and our framework can be generalized to other music annotations (e.g., emotion, instrument, etc.). Furthermore, our method enables fine-grained control over how much the retrieval process focuses on self-supervised vs. label information at inference time. We evaluate the learned embeddings through a variety of video-to-music and music-to-video retrieval tasks. Our experiments show that the proposed approach successfully combines self-supervised and supervised objectives and is effective for controllable music-video retrieval.