NEW DELHI — When President Obama visited India in 2010, he called the warming relationship between it and the United States the “defining partnership of the 21st century.” Decades of disagreements, from Cold War ideological battles to squabbles over the United States’ close relationship with India’s archrival, Pakistan, would take a back seat to the many shared interests of two of the world’s largest and most diverse democracies.

But almost four years later, the United States and India have found themselves on opposite sides of the world’s most important diplomatic issues, from the crisis in Ukraine, in which India came to Russia’s defense, to a long-awaited vote to investigate Sri Lanka’s government for atrocities committed at the end of its civil war (India abstained). Even critical military coordination over the reduction of troops in nearby Afghanistan has suffered.

Far from coordinating on major global issues, the two countries are embroiled in a series of spats over privileges, visas and even swimming pools in a nasty fight stemming from the arrest and strip-search in New York City of Devyani Khobragade, an Indian consular official, in December on charges of submitting false documents to obtain a work visa for a housekeeper whom she then severely underpaid.

The arrest infuriated senior members of India’s diplomatic service, many of whom had paid their maids comparably when posted in New York, a plum assignment. For them, the arrest was one of a series of American actions deemed insensitive here.