HONG KONG — China and India have jockeyed for centuries over the Himalayas. The Chinese military invaded Tibet in 1950. India granted asylum to the Dalai Lama, the Tibetan spiritual leader, in 1959. Three years later, the two countries fought a border war. Now they are in a standoff over an area disputed by China and Bhutan, the Himalayan kingdom whose claim is supported by India.

The two countries’ latest struggle is over which one will be able to formally tie the ancient practice of Tibetan medicine to its national patrimony. The prize: international cachet and the possibility of significant commercial rewards.

In March, China filed paperwork asking the United Nations Educational, Scientific and Cultural Organization to recognize medicinal bathing, one of many practices of sowa rigpa, the Tibetan name for this type of medicine, as part of its “intangible cultural heritage.” Unesco’s website indicates that the request will be considered next year. India filed its own bid — for the entire sowa rigpa tradition — around the same time.

“If China is applying, of course India can also apply,” said Geshe Ngawang Samten, the vice chancellor of the Central University of Tibetan Studies in Sarnath, India. “This is Indian culture as well.”