I will try to keep this relatively vague in an attempt to be respectful to the people who were involved...
My feeling is that if your horse is a polite and respectful animal on the ground, then your work and the respect you've obtained should translate properly under saddle.
So today I went to spend some time with friends at another barn. This lady offered her horse to me for the day. She's been through some "stuff" and struggles with her confidence and her trust in her horse.
He didn't concern me in the least. He was rude and there were things that, were he my horse, I would address them IMMEDIATELY. But I worked him on the ground before I got in the saddle, and he was decent to ride around the property and in the arena.
The trouble happens when the owner decides that she wants to get on. She was so happy with how he was with me, that she thought (after 2 months of not riding) that she would be able to get on and go. That lasted about 4 seconds. She mounted from the block, he wheeled around, and bolted for his stall. She had to jump off to avoid being decapitated by the stall roofing!!
So here's my question:
This horse in particular was trained in a specific form of NH and has been since birth. However, my feeling is that although he's been very well ground trained, he was never properly under saddle trained. But wouldn't you suppose that a horse that has a solid foundation of ground training, should be much more polite and respectful under saddle as well?! I'm at a loss here.