I started Straightness Training without longe line recently. In the video Romanesque, El Blanco, Prince Elmelund and Toronto are really forthcoming, and while they are in total liberty they are trying their best. On the circle I am looking for lateral bending, a forward down tendency, and stepping under with the inside hindleg. It is just work in progress, but the beginning is there:
You don't always need tools or physical aids to get the horse to do what you want. Your inner picture, energy and body language are very effective, and they are your most important means of communication. The whip can be seen as a tool to lengthen your arm and your energy, and not as a physical aid where you actually touch the horse.
With your inner picture, feeling and body language you can control all four legs of the horse!
Your horse is constantly observing you, and he can read your mind, feel your energy and see your body language at any time. Therefore, they are very effective and important means of communication.
Romanesque sometimes does the 'left' terre à terre on the right side, and he also does a 'cross' terre à terre sometimes, just like in canter you can get a cross-canter or a 'wrong' canter.
So even in the terre à terre you need to straighten your horse ;). And most of all: "Ride the horse forward and straight", so by asking the horse to go a tiny bit forward in a right schoolcanter, and then collecting him again into terre à terre, you can keep the terre à terre 'clean'.