The Collaborative International Dictionary of English v.0.48:

Bearing rein \Bear"ing rein`\
   A short rein looped over the check hook or the hames to keep
   the horse's head up; -- called in the United States a
   checkrein.
   [1913 Webster]

The Collaborative International Dictionary of English v.0.48:

Checkrein \Check"rein`\, n.
   1. A short rein looped over the check hook to prevent a horse
      from lowering his head; -- called also a bearing rein.
      [1913 Webster]

   2. A branch rein connecting the driving rein of one horse of
      a span or pair with the bit of the other horse.
      [1913 Webster]

WordNet (r) 3.0 (2006):

bearing rein
    n 1: a rein designed to keep the horse's head in the desired
         position [syn: bearing rein, checkrein]