Autonomous artificial intelligence (AI) agents are increasingly deployed on blockchain platforms,  yet the design space governing their interaction remains poorly understood. This convergence, in which autonomous agents operate on and within decentralized systems, characterizes the emerging Web 4.0  paradigm. We organize this Systematization of Knowledge (SoK) around a bidirectional trust framework.  For the B → A direction (Blockchain → Agent), blockchain provides the trust infrastructure needed by agents. This direction follows their on-chain lifecycle: identity and account abstraction establish an agent on-chain; permission and delegation define what it may do; intent-centric execution carries out its goals; and tokenized agent economies support economic participation. The A → B direction (Agent → Blockchain) concerns participation in core blockchain mechanisms, beginning with security auditing and extending to consensus and governance. Verifiable computation forms the Trust Foundation (TF) shared by both directions. We consider zero-knowledge machine learning (zkML) and optimistic machine learning (opML) alongside trusted execution environments (TEEs). Their trade-offs between trust minimality and computational overhead differ, as does deployment readiness. The Agent–Blockchain Interaction Model (ABIM) formalizes this interaction. We catalog 70 Ethereum Improvement Proposals (EIPs) and Ethereum Request for Comments (ERC) standards in the Appendix and review 127 academic papers. The analysis also covers 20 representative industry projects. This material is compared using five dimensions: Verifiability, Minimality of Trust, Expressiveness, Composability, and Maturity. The comparison reveals three unresolved issues. The agent-specific standards ecosystem is predominantly immature, with only 3 of 13 direct AI/agent  ERCs having reached Final status. Intent architectures lack formal analysis. Research on AI participation in consensus and governance remains limited to isolated studies, and a unified security framing that treats  AI as a first-class actor at the protocol layer is still absent. We propose a three-dimensional taxonomy of agent autonomy, trust model, and operational direction. We identify nine concrete open problems and outline the principal research opportunities.