Propaganda and persuasion technique detection, the task of identifying how misleading content persuades rather than merely what it claims, occupies a distinctive and underexplored position within computational misinformation research. This task requires interpretable and rhetorically grounded reasoning over persuasion techniques, making it particularly well aligned with neurosymbolic artificial intelligence (AI). This survey presents the first systematic review of neurosymbolic approaches for propaganda and persuasion technique detection, together with relevant structured-reasoning approaches from closely related areas that offer methodological foundations for these tasks. It synthesizes work spanning argument mining, knowledge-augmented learning, logic-based fallacy detection, structured reasoning with large language models (LLMs), and hybrid neural-symbolic architectures. We further organize existing methods through a unified taxonomy of neurosymbolic integration paradigms and review foundational rhetoric theory, benchmark datasets, and evaluation methodologies. We identify key open challenges, including subtle and implicit persuasion, cross-lingual generalization, explanation faithfulness, and the growing challenge of LLM-generated propaganda. Finally, we outline future research directions and argue that neurosymbolic AI provides a promising foundation for bridging predictive performance with human-interpretable rhetorical reasoning in next-generation misinformation detection systems.