Self-supervised speech representation models have succeeded in various tasks, but improving them for content-related problems using unlabeled data is challenging. We propose speaker-invariant clustering (Spin), a novel self-supervised learning method that clusters speech representations and performs swapped prediction between the original and speaker-perturbed utterances. Spin disentangles speaker information and preserves content representations with just 45 minutes of fine-tuning on a single GPU. Spin improves pre-trained networks and outperforms prior methods in speech recognition and acoustic unit discovery.