With the mainstream integration of machine learning into security-sensitive domains such as healthcare and finance, concerns about data privacy have intensified. Conventional artificial neural networks (ANNs) have been found vulnerable to several attacks that can leak sensitive data. Particularly, model inversion (MI) attacks enable the reconstruction of data samples that have been used to train the model. Neuromorphic architectures have emerged as a paradigm shift in neural computing, enabling asynchronous and energy-efficient computation. However, little to no existing work has investigated the privacy of neuromorphic architectures against model inversion. Our study is motivated by the intuition that the non-differentiable aspect of spiking neural networks (SNNs) might result in inherent privacy-preserving properties, especially against gradient-based attacks. To investigate this hypothesis, we propose a thorough exploration of SNNs' privacy-preserving capabilities. Specifically, we develop novel inversion attack strategies that are comprehensively designed to target SNNs, offering a comparative analysis with their conventional ANN counterparts. Our experiments, conducted on diverse event-based and static datasets, demonstrate the effectiveness of the proposed attack strategies and therefore questions the assumption of inherent privacy-preserving in neuromorphic architectures.