Sarah, if your girl does those things, then it would suggest to me, sitting thousands of miles away, that maybe she is indeed nervous of 'the cage' - or that she remembers something that happened to her whilst she was in the cage.
You'll have to be persistent and keep working with her - but I would suggest on a one-to-one basis only - and never get angry with her.
In Le Trec training, there are a number of exercises designed to train a rider interested in trail riding - including map reading from horseback.
One of the tests is to set out on the ground two lines of long poles, laid down parallel to each other, a horse width apart, forming a right angle. The horse, mounted by its rider has to back up and turn the right angle without touching the poles. It is quite a difficult test.
To get the horse to perform this move usually calls for training in hand from the ground - the horse must remain steady, take one step at a time and be prepared to put its feet where it cannot see. The trainer uses a lead rein off a head collar, a long light dressage whip (to lightly tap with and never to hit with) and the pointed finger to the chest.
From the saddle the movement is slightly easier to perform, because the horse should then know what is being asked of it and will perform upon being given the correct aid and command.
In Hilda Nielson's book entitled "Francois Baucher" ISBN 10 0-85131-534-8 she states on page 89 that Baucher a 19th century contraversial horsemaster invented 'the backward trot' . Why a horse should be asked to trot backwards defeats me, but this man was very clever with horses and his aim in life was to illustrate to the French aristocracy and gentry what could be done with them.
So getting your girl to back into that 'infernal' cage should be easily possible if you take the time to show her what you want her to do. Remember the finger; forget assistants, shouting and any aggression on your part. Just ask with persistance.
Best of luck