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Spiking neural networks (SNNs) have attracted much attention for their high energy efficiency and for recent advances in their classification performance. However, unlike traditional deep learning approaches, the analysis and study of the robustness of SNNs to adversarial examples remain relatively underdeveloped. In this work, we focus on advancing the adversarial attack side of SNNs and make three major contributions. First, we show that successful white-box adversarial attacks on SNNs are highly dependent on the underlying surrogate gradient technique, even in the case of adversarially trained SNNs. Second, using the best surrogate gradient technique, we analyze the transferability of adversarial attacks on SNNs and other state-of-the-art architectures like Vision Transformers (ViTs) and Big Transfer Convolutional Neural Networks (CNNs). We demonstrate that the adversarial examples created by non-SNN architectures are not misclassified often by SNNs. Third, due to the lack of an ubiquitous white-box attack that is effective across both the SNN and CNN/ViT domains, we develop a new white-box attack, the Auto Self-Attention Gradient Attack (Auto-SAGA). Our novel attack generates adversarial examples capable of fooling both SNN and non-SNN models simultaneously. Auto-SAGA is as much as $91.1\%$ more effective on SNN/ViT model ensembles and provides a $3\times$ boost in attack effectiveness on adversarially trained SNN ensembles compared to conventional white-box attacks like Auto-PGD. Our experiments and analyses are broad and rigorous covering three datasets (CIFAR-10, CIFAR-100 and ImageNet), five different white-box attacks and nineteen classifier models (seven for each CIFAR dataset and five models for ImageNet).