It has been widely observed that language models (LMs) respond in predictable ways to algorithmically generated prompts that are seemingly unintelligible. This is both a sign that we lack a full understanding of how LMs work, and a practical challenge, because opaqueness can be exploited for harmful uses of LMs, such as jailbreaking. We present the first thorough analysis of opaque machine-generated prompts, or autoprompts, pertaining to 6 LMs of different sizes and families. We find that machine-generated prompts are characterized by a last token that is often intelligible and strongly affects the generation. A small but consistent proportion of the previous tokens are prunable, probably appearing in the prompt as a by-product of the fact that the optimization process fixes the number of tokens. The remaining tokens fall into two categories: filler tokens, which can be replaced with semantically unrelated substitutes, and keywords, that tend to have at least a loose semantic relation with the generation, although they do not engage in well-formed syntactic relations with it. Additionally, human experts can reliably identify the most influential tokens in an autoprompt a posteriori, suggesting these prompts are not entirely opaque. Finally, some of the ablations we applied to autoprompts yield similar effects in natural language inputs, suggesting that autoprompts emerge naturally from the way LMs process linguistic inputs in general.