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Time-series foundation models have the ability to run inference, mainly forecasting, on any type of time series data, thanks to the informative representations comprising waveform features. Wearable sensing data, on the other hand, contain more variability in both patterns and frequency bands of interest and generally emphasize more on the ability to infer healthcare-related outcomes. The main challenge of crafting a foundation model for wearable sensing physiological signals is to learn generalizable representations that support efficient adaptation across heterogeneous sensing configurations and applications. In this work, we propose NormWear, a step toward such a foundation model, aiming to extract generalized and informative wearable sensing representations. NormWear has been pretrained on a large set of physiological signals, including PPG, ECG, EEG, GSR, and IMU, from various public resources. For a holistic assessment, we perform downstream evaluation on 11 public wearable sensing datasets, spanning 18 applications in the areas of mental health, body state inference, biomarker estimations, and disease risk evaluations. We demonstrate that NormWear achieves a better performance improvement over competitive baselines in general time series foundation modeling. In addition, leveraging a novel representation-alignment-match-based method, we align physiological signals embeddings with text embeddings. This alignment enables our proposed foundation model to perform zero-shot inference, allowing it to generalize to previously unseen wearable signal-based health applications. Finally, we perform nonlinear dynamic analysis on the waveform features extracted by the model at each intermediate layer. This analysis quantifies the model's internal processes, offering clear insights into its behavior and fostering greater trust in its inferences among end users.