It’s true that Robespierre went to the guillotine with a heavily bandaged face, and that this was removed before his execution. He had been shot - possibly by his own hand - the day before, and shattered his jaw. He would have been in a great deal of pain; eyewitness accounts describe him bleeding heavily when the bandages and splinting that held his face together were removed.

Here’s one

It would be difficult to describe the appearance of Robespierre. His face was wrapped in a bandage of dirty, bloodstained linen; his features were horribly disfigured. A pale pallor made it even more repulsive. He kept his eyes downward and almost closed; whether that was from the pain of his wounds or an awareness of his misdeeds, one cannot say.

Just before arriving at the place of execution, Robespierre was shaken out of his lethargy by a woman who forced her way through the crowd and rushed up to the cart carrying this cannibal. She grasped the cart rail with one hand and threatened Robespierre with the other, saying “Monster spewed from Hell. The thought of your punishment intoxicates me with joy”. Robespierre opened his eyes and looked at her sadly as she said: “Go now, evildoer, go down into your grave carrying the curses of the wives and mothers of France!”

When the cart had reached the foot of the scaffold, the executioners carried the tyrant down and laid him out prone until it was his turn for execution. While his accomplices were being beheaded, Robespierre appeared not to be taking notice; he kept his eyes shut and did not open them until he himself was carried up the scaffold. Some said that when he saw the instrument of death, he heaved a sigh of pain, but before dying he had to endure bitter suffering. After having thrown down his coat, the executioner roughly tore away the bandage and splint which the surgeon had put on his wounds. This unhinged his lower jaw from the upper jaw and caused the blood to flow in torrents.

However, removing the supports was probably not done on purpose to cause him pain. It clearly did hurt him, but the pain is unlikely to have been the point.

The whole point of the guillotine as an execution device was that it was the “humane” way - relatively quick, relatively clean compared to the other options French law allowed - and the executioner Charles Henri Sanson was well known as a conscientious, professional man. He did not, as a rule, tolerate shenanigans, petty cruelties or disrespect when he was working; during the execution of Charlotte Corday at the height of the Terror, Sanson was vocally indignant at the suggestion that one of his assistants had mistreated or disrespected her head by picking it up and slapping her. In as far as one can have pride in his work as an executioner, Sanson seems to have done so.

He would have removed the bandages and splinting on Robespierre for the same reason he had cropped Louis XVI’s hair and taken away his outer clothing before HIS death - the blade of the guillotine, as it swung down, needed as few obstacles in the way as possible for a clean severing cut. If the blade had got stuck...

The evidence for face up vs face down is less clear. We know that face down executions were more common, but pictorial depictions of Thermidor differ in what they show for Robespierre specifically. Some images show him face down, others repeat the rumour that he died screaming face up. He would have seen and been aware of the blade before being strapped in in either scenario.

It may have depended on how that specific guillotine was set up.