Don't think they refuse to play each other ... for Reidsville its tough to find non-conference match-ups on the dates available when they are locked into three in-county games. Northeast Guilford and Western Alamance are both closer than Dudley and they are not on the schedule either. Western Alamance was always a huge gate when Reidsville played them.
It wasn't too many seasons ago when Dudley vs Carver was a huge rivalry game as the Durham Hillside game is now. The other three non-conference games on Dudley's non-conference schedule are Guilford County foes.