The key question is whether America and China will be increasingly acrimonious competitors or cooperative partners, albeit with very different political systems.
Will China's president wag his finger or lend a hand in Washington?
Hu Jintao will travel to the United States next week for his third official visit as the Chinese president. It may be his last before he hands over power to his apparently designated successor, the Chinese vice president Xi Jinping, in 2012 - coincidentally the same year that the US president Barack Obama is likely to be campaigning for a second term in the White House.
According to Forbes magazine, Mr Hu is the most powerful man in the world. Leaving aside the fact that power at the top is much more bureaucratically institutionalised in China than it was in Mao Zedong's day (a good thing), certainly this visit is hugely important. Indeed, the US-China relationship will be the most significant bilateral engagement in shaping the course of the 21st century.
At the heart of globalisation has been the emergence of fast-growing economies, most notably Brazil, India, and, above all, China. The US, of course, remains the world's only superpower - militarily, economically, politically, and culturally. While the world's democracies are not slow to criticise American leadership, they know that they rely on the US in tackling the most serious global problems. Without America, nothing much gets done.
But China now has enough commercial clout, backed by more than $2 trillion in foreign-exchange reserves, to play a decisive role in advancing or impeding global problem-solving, from the G-20 agenda to efforts to rein in North Korea's nuclear ambitions. China is far too big to be taken for granted, and it wants to be shown the respect that it associates with being an ancient civilisation that has contributed so much to human progress.
For the rest of us, the key question is whether America and China will be increasingly acrimonious competitors or cooperative partners, albeit with very different political systems. Will they fight to dominate the century, or to make it more peaceful and prosperous?
China has become surprisingly maladroit in handling the US and its Asian neighbours in recent months. Its leaders seem to have interpreted Mr Obama's attempts to engage with them, down-playing bilateral aggravations, as a sign of American weakness in the wake of Wall Street's crash and military reverses in Iraq and Afghanistan. Arrogance has replaced sophisticated modesty. What else can explain the treatment of Mr Obama on his first trip to China and during the disastrous 2009 climate-change conference in Copenhagen, where a relatively minor Chinese official wagged his finger in the face of the US president?
China's official behaviour following the award of the Nobel Peace Prize to Liu Xiaobo turned an embarrassment into a public-diplomacy disaster, and China's neighbours have been disturbed by Chinese efforts to throw its weight around. Japan, Vietnam, and even Singapore have reacted with consternation, highlighting the need for America to remain the principal guarantor of stability in Asia.
It is surprising that this has happened under Mr Hu, a cautious and intelligent man. Maybe this behaviour is attributable to the imminent leadership change, with an aggressive faction in the Communist Party Politburo and Central Committee needing to be mollified. There must be some explanation for China choosing this moment even for an unnecessary and ham-fisted row with the Vatican.
So the stakes in Washington are high for Mr Hu. He will hear for himself the strength of American arguments about trade and the renminbi's exchange rate. He will be able to point out, at least in private, that if you look at the real effective exchange rate - taking account of the impact on export prices of rising labour costs - the renminbi-dollar gap is a lot less important than China's critics suggest.
But he must also provide some real evidence that China is opening its markets as domestic consumption grows, and that it recognises that a sustainable global recovery requires adjustments in China as well as in America to redress international imbalances.
On the security front, China should show that it shares the nervousness in America, Europe, and the Middle East about the nuclear ambitions of North Korea and Iran. It is not enough to hope for the best. No one doubts that North Korea is responsible for its own delinquent behaviour. But China's public failure to distance itself from the North's military provocations has undermined its credibility in efforts to defuse the crisis.
More importantly, China must make clear that it will support tougher sanctions on Iran - and help to implement them - if the Iranian regime continues to lie about its nuclear programme. Iran's oil and gas should not blind China to the dangers of its neighbourhood and the entire world if the Islamic Republic develops a nuclear weapon.
China deserves to be treated seriously as a major player in global governance. But, in order to secure the status that it desires, it must demonstrate that it understands that partnership is a two-way street.
Chris Patten, the last British governor of Hong Kong and a former EU commissioner for external affairs, is Chancellor of the University of Oxford
© Project Syndicate 2011