Federated clustering addresses the critical challenge of extracting patterns from decentralized, unlabeled data. However, it is hampered by the flaw that current approaches are forced to accept a compromise between performance and privacy: \textit{transmitting embedding representations risks sensitive data leakage, while sharing only abstract cluster prototypes leads to diminished model accuracy}. To resolve this dilemma, we propose Structural Privacy-Preserving Federated Graph Clustering (SPP-FGC), a novel algorithm that innovatively leverages local structural graphs as the primary medium for privacy-preserving knowledge sharing, thus moving beyond the limitations of conventional techniques. Our framework operates on a clear client-server logic; on the client-side, each participant constructs a private structural graph that captures intrinsic data relationships, which the server then securely aggregates and aligns to form a comprehensive global graph from which a unified clustering structure is derived. The framework offers two distinct modes to suit different needs. SPP-FGC is designed as an efficient one-shot method that completes its task in a single communication round, ideal for rapid analysis. For more complex, unstructured data like images, SPP-FGC+ employs an iterative process where clients and the server collaboratively refine feature representations to achieve superior downstream performance. Extensive experiments demonstrate that our framework achieves state-of-the-art performance, improving clustering accuracy by up to 10\% (NMI) over federated baselines while maintaining provable privacy guarantees.