Functional Connectivity Representation (FCR) free of brain parcellations, such as diffusion maps of functional connectivity, provides critical insights into brain functional organization, making it essential in neuroimaging studies. However, site-specific variability in multi-site fMRI datasets caused by scanner and imaging protocol differences introduces non-biological variability, necessitating harmonization for reproducible and generalizable cross-cohort studies. Despite existing efforts, previous methods typically overlook some critical properties during harmonization, such as local functional, longitudinal, and population-affinity relation, limiting the reliability and interpretability of downstream analyses. Therefore, we propose Relation-Preserving Functional Connectivity Representation Harmonization (RP-FCRH), a novel framework that removes site effects while preserving biologically meaningful variability in FCR. For site effect removal, RP-FCRH is built upon a Cycle-Consistent Adversarial Autoencoder (CAA), which enforces site-invariance in the latent space via a discriminator and utilizes cycle consistency to generate realistic predictions. To preserve biological variability, we introduce three relation-preserving constraints based on CAA: (1) Local Functional Stability: RP-FCRH maintains the spatial organization of local functional connectivity patterns by preserving local gradient structures, ensuring fine-grained functional relationships remain intact. (2) Longitudinal Trend Consistency: RP-FCRH preserves individual-specific developmental trajectories by minimizing deviations in longitudinal trajectories before and after harmonization. (3) Population-Level Similarity: RP-FCRH maintains similarity relationships among individuals within a cohort by constraining inter-subject distances, preventing artificial alterations in within-group characteristics. Extensive experiments on 4 fMRI datasets (1,206 scans) highlight the superiority of RP-FCRH in reducing site-specific variability while preserving critical functional connectivity relationships, demonstrating its potential in enabling more robust and generalizable cross-cohort fMRI studies.