While we see the known / usual suspects, Indian history is rich in such rivalries. One such rivalry was between Emperor Pulakesin II of the Chalukya Empire and King, later Emperor Narasimhavarman I of the Pallava Empire.
Pulakesin I who commanded an empire roughly the size of Modern France, Germany, Spain and Italy combined, turned his attention to the South and attacked the Vishnukundis a tiny kingdom, but an ally of the smaller (compared to the Chalukyas) Pallava Kingdom. The then tiny Pallava Empire (commanding an area roughly equal to Modern day Austria) declared war on the Chalukyas.
The King of the Pallava Kingdom then was the erudite, literature and arts loving King Mahendravarman I, he lead an able resistance, fighting and winning a series of running battles with the Chalukyans. At some point in time, Mahendravarman had to make a choice, retreat to his capital and save it or spread himself thin and lose it all. So after inflicting a crushing defeat on a Chalukyan army at a place called Pullalur, retreated just in time to fight off a siege of his capital Kanchi, but his and his armies strength was spent. The Chalukyas ravaged the rest of the kingdom and the South of India, made vassals of other smaller kingdoms, extracted tribute from Mahendravarman and left. Heartbroken, Mahendravarman retreated into a shell, pouring all his energies into building beautiful temples, constructing highways etc etc. He is said to have died of sadness 10 years after the above events.
Enter his son Narasimhavarman I. Tamil lore says that his father made him promise eternal vengeance and revenge on Pulakesin II. It is also entirely possible that Narasimhavarman I fought as a general in his fathers wars against the Chalukyas, and thus was already an able and experienced soldier when he took charge of the Pallava Kingdom.
Upon taking the throne, Narasihavarman set about building a coalition to take on the Chalukyan power which had expanded even more and was at that point in time, the pre-eminent power in the Indian subcontinent and then some. His coalition consisted of Sinhalese (Sri Lankan) troops under Prince Manavarma, the Pandyan kingdom and interestingly enough, a warrior monk named Paranjothi. Paranjothi was a foot soldier who had caught the eye of Mahendravarman and was promoted up the ranks. He was to become the Indian Belisarius.
Round II - As Narasimhavarman expected, after defeating North Indian powers, Pulakesin II wanted to complete what he had started some 15 years ago, the total subjugation of the tiny holdout Pallava Kingdom.
After a few non consequential battles, Narasihavarman took to the field himself, and faced his arch enemy Pulakesin II at the battle of Manimangalam. Resounding defeat for Pulakesin II who was forced to retreat back to his lands. The Pallavas, drunk on success, followed the retreating Chalukyan army right into their capital. Pulakesin gathered in all his troops from the Northern and Eastern borders for one last throw of the die.
In the battle of Vatapi (Badami) he lost, and legend has it that he and Narasimhavarman met in combat mano a mano and Narasimhavarman killed his arch enemy in single combat.
Narasimhavarman then burnt the city of Vatapi to the ground (or at least allowed a major sack) and thus ended the rivalry between Pulakesin and Narasihmavarman.