Spatial joins are among the most time-consuming spatial queries, remaining costly even in parallel and distributed systems. In this paper, we explore hardware acceleration for spatial joins by proposing SwiftSpatial, an FPGA-based accelerator that can be deployed in data centers and at the edge. SwiftSpatial contains multiple high-performance join units with innovative hybrid parallelism, several efficient memory management units, and an extensible on-chip join scheduler that supports the popular R-tree synchronous traversal and partition-based spatial-merge (PBSM) algorithms. Benchmarked against various CPU and GPU-based spatial data processing systems, SwiftSpatial demonstrates a latency reduction of up to 41.03x relative to the best-performing baseline, while requiring 6.16x less power. The performance and energy efficiency of SwiftSpatial demonstrate its potential to be used in a variety of configurations (e.g., as an accelerator, near storage, in-network) as well as on different devices (e.g., data centers where FPGAs are widely available or mobile devices, which also contain FPGAs for specialized processing).