HollyBubbles, there is no problem with a horse turning around to face backwards in the trailer. Provided once there, (s)he stays there and doesn't move around as in kitten_Val's example.
My Lisa much prefers to ride backwards. I still have to see if she will fit backwards in one stall in my new trailer. I hope so; then she can be happy facing backwards and my friend's horse can come with us in the other side. When left to her own devices, Lisa will face the back every time.
In answer to the OP, I much prefer to leave my horse untied. If your horse is quiet it is safer for the horse. They can find their own balance that way and if, heaven forbid, there is a slip and fall, or an accident, there is less risk of injury to the neck.
Here's a question -- what about two horses, untied, without a divider in a two horse straight load goose neck? Assuming the two horses get along, of course. Anyone ever done that?