A dual-cam linkage system controls the orientation of tines on a reel of a crop harvesting header. The dual-cam linkage system includes a stationary, substantially circular first cam track that has a rotational center that is offset from the rotational axis of the reel. A second cam track is located in a support plate on which a tine-supporting member is mounted. The dual-cam linkage system also includes a pivoting linkage having a crank, an arm. The crank has a first end fixedly attached to the tine-supporting member. The arm has an outer end attached to a second end of the crank. The first cam follower is attached at an inner end of the arm and engages the first cam track. The second cam follower is attached to the arm at a position between the first cam follower and the crank and engages the second cam track.