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Self-supervised learning (SSL) models, such as Wav2Vec2, HuBERT, and WavLM, have become foundational across a wide range of speech and audio tasks. Despite their success, understanding their internal layer-wise dynamics remains an ongoing challenge. To address this, we propose a two-part model-centric framework called INSIDESSL. First, we establish a task-agnostic analysis from three intrinsic per-layer perspectives: compression (entropy), geometry (curvature), and robustness to perturbations. We show that varying training objectives induce distinct regimes of acoustic compression and manifold unfolding. Second, we introduce the cross-layer Generative Compatibility Matrix (GCM) to evaluate functional transferability, exposing stable phonetic cores, identity volatility, and deep-layer semantic pruning. In addition to these evaluations, linear probing connects the model-centric perspective to downstream tasks, demonstrating how layer topology dictates phoneme, pitch, and speaker encoding.