When Prime Minister Narendra Modi, let down by his once “true friend” President Trump over choking tariffs on India, clasped hands this week with the leaders of Russia and China, he was clearly trying to send a message.

Leaving no chance to the official photographers, Mr. Modi’s interpreter pulled out his phone and circled the three leaders for close-ups.

“They were hoping I was watching,” remarked Mr. Trump, whose administration has been unraveling two decades of American courtship of India. “And I was watching.”

But if Mr. Modi’s moment of unity with U.S. rivals sent one message, his absence from photographs the next day, when dozens of leaders attended a huge Chinese military parade, sent another.

Mr. Modi had quickly jetted back to New Delhi, skipping a spectacle that would remind voters back home that some of China’s military hardware is pointed at India.

India’s foreign policy has long been a difficult balancing act of nonalignment — relations and deals in all directions, without getting too close to any one country at the cost of the other.