Did the Centaurs participate in the Battle of Hogwarts?

Question

Centaurs are the magical creatures lived in the Forbidden Forest, at Hogwarts School of Witchcraft and Wizardry. In Sorcerer's Stone a centaur, named Firenze who rescued Harry in the Forbidden Forest and offers to give him a ride back to the school. Did the Centaurs participate in the Battle of Hogwarts and whose side were they on?

Answer 1

Not initially, but they joined the fight before the end.

When Hagrid is bringing Harry back from the Forest, he’s seen by a procession of centaurs, and Hagrid makes it clear that they didn’t fight:

“BANE!”

Hagrid’s unexpected bellow nearly forced Harry’s eyes open. “Happy now, are yeh, that yeh didn’ fight, yeh cowardly bunch o’ nags? Are yeh happy Harry Potter’s — d-dead…?”

Hagrid could not continue, but broke down in fresh tears. Harry wondered how many centaurs were watching their procession pass; he dared not open his eyes to look. Some of the Death Eaters called insults at the centaurs as they left them behind. A little later, Harry sensed, by a freshening of the air, that they had reached the edge of the forest.

— Deathly Hallows, chapter 36 (The Flaw in the Plan)

Honourable mention must go to Firenze, who did join the fight. As a Hogwarts teacher and outcast from the Hogwarts centaur herd, he was motivated more by his duty to the school and students, than to the whims of the herd:

Behind her stood the remaining teachers, including the palomino centaur, Firenze, and the members of the Order of the Phoenix who had arrived to fight.

— Deathly Hallows, chapter 31 (The Battle of Hogwarts)

Hagrid’s speech clearly motivates them to action, because when Hagrid reaches the castle and Voldemort is monologuing, the centaurs charge the castle (accompanied by Grawp). Here are a few excerpts from the book that mention centaur action:

They heard uproar from the distant boundary of the school as what sounded like hundreds of people came swarming over the out-of-sight walls and pelted toward the castle, uttering loud war cries. […]

The slash of the silver blade could not be heard over the roar of the oncoming crowd or the sounds of the clashing giants or of the stampeding centaurs, and yet it seemed to draw every eye. […]

Chaos reigned. The charging centaurs were scattering the Death Eaters, everyone was fleeing the giants’ stamping feet, and nearer and nearer thundered the reinforcements that had come from who knew where. […]

The centaurs Bane, Ronan, and Magorian burst into the hall with a great clatter of hooves, as behind Harry the door that led to the kitchens was blasted off its hinges.

— Deathly Hallows, chapter 36 (The Flaw in the Plan)

It includes the three named centaurs from the books, and Bane, who Hagrid called out earlier. So they fought for the defenders of Hogwarts, but initially kept themselves out of it until Harry was (apparently) killed.

Answer 2

According to an exchange between Harry Potter and Bane in Harry Potter and the Cursed Child, not only did the Centaurs fight bravely, but they were gifted the Forbidden Forest as reward for their timely intervention.

HARRY: I have always respected the centaurs. We are not enemies. You fought bravely at the Battle of Hogwarts. And I fought beside you.

BANE: I did my part. But for my herd, and our honor. Not for you. And after the battle, the forest was deemed centaur land. And if you’re on our land — without permission — then you are our enemy.