The Generate Near Table tool will create duplicate pairs (IN_FID-NEAR_FID and NEAR_FID-IN_FID) in the Near Table when using the "Find only closest feature" option. It would be nice if the tool offered the option (e.g., a checkbox) to eliminate pairs - keep one, toss one. Non-duplicated distances are often desirable for statistical purposes. You could use the Delete Identical tool afterward to eliminate duplicate distances, but that is not a good option because there can be legitimate instances of identical distances! Zero is a particularly likely distance of which there may be many legitimate (non-paired) instances. It is possible to use SQL to find duplicate pairs in separate columns regardless of column order (IN_FID and NEAR_FID), but that is pretty advanced, particularly within the SQL limitations of ArcGIS.