In this paper, we propose a novel self-supervised transfer learning method called \underline{\textbf{D}}istribution \underline{\textbf{M}}atching (DM), which drives the representation distribution toward a predefined reference distribution while preserving augmentation invariance. DM results in a learned representation space that is intuitively structured and therefore easy to interpret. Experimental results across multiple real-world datasets and evaluation metrics demonstrate that DM performs competitively on target classification tasks compared to existing self-supervised transfer learning methods. Additionally, we provide robust theoretical guarantees for DM, including a population theorem and an end-to-end sample theorem. The population theorem bridges the gap between the self-supervised learning task and target classification accuracy, while the sample theorem shows that, even with a limited number of samples from the target domain, DM can deliver exceptional classification performance, provided the unlabeled sample size is sufficiently large.