If a gag bit, there will be a drop noseband in addition to the cavesson supporting the tie-down.
From the en.wikipedia.org
Generally the bridle is open-faced, without a noseband, unless the horse is ridden with a tiedown.
From the en.wikipedia.org
At the lower levels of dressage, a bridle includes a plain cavesson, drop noseband, or flash noseband.
From the en.wikipedia.org
Later, as Imperial Commander took the ovation, Walsh could be seen standing up in the saddle as Kauto Star's white noseband approached through the gloom.
From the guardian.co.uk
More examples
A strap that is the part of a bridle that goes over the animal's nose
A noseband is the part of a horse's bridle that encircles the nose and jaw of the horse. In English riding, where the noseband is separately attached to its own headstall (headpiece), it is often called a cavesson or caveson. ...
The noseband is that part of a bridle that fits around the horse's face and buckles under the jaw bars which encourages the horse from opening his mouth.