Pierre Trudeau, the urbane and contrarian Canadian prime minister, once said living next to the United States was like being a mouse sleeping with an elephant: cozy, yes, but if the elephant rolls over carelessly, the consequences are huge.

The mouse visits the elephant in 10 days, when Stephen Harper, the heir to Mr. Trudeau's office, and the Canadian delegation arrive here for the Group of 20 Summit. Canada's cultural and political ties are so close that a standard joke has it that a Canadian is a polite, unarmed American with health insurance.

That politeness masks a far more complex political and economic relationship that sometimes strains over issues barely on the radar of average Americans: softwood lumber tariffs, intellectual property rights and diversion of rivers.

And there is health care. Canada's national health insurance program has become the bugaboo invoked by opponents of American health care reform, with one scary commercial featuring a Canadian woman who says she was told to wait for life-saving treatment for a brain tumor. (Canadian officials say she had a cyst that was not life-threatening and that she was scheduled for surgery in plenty of time to avoid complications.)

"Not that there isn't an awkwardness between Harper and Obama waiting to happen," said Patrick James, a Canadian citizen and professor of international relations at the University of Southern California.

That awkwardness could come, in part, from Canada's reluctance to have its health care system demonized for the benefit of American politicians. Too, there is nettlesome "buy American" language in the federal stimulus bill that could shut out Canadian firms from trade based on government subsidized economic growth here.

Above all, there is, says Mr. James, the ever-present worry that any Canadian leader faces: being seen as a sidekick.

"There is always a problem with Canadian prime ministers -- you can't seem to be a lackey of the American president," Mr. James said.

Nor does it help to be seen as a rival.

Mr. Trudeau courted trouble in the 1970s when he talked about the so-called "third option" of seeking trade and relationships outside the U.S. orbit. It went nowhere, if only because of growing bonds between the two nations that included the 1965 "Auto Pact" that helped curb tariffs, created a single market for automobile manufacturing and spurred the opening of Canadian plants by the Big Three automakers.

Today, parts for the average car cross the U.S.-Canada border a dozen times before it rolls off the assembly line, be it in Michigan or Ontario.

While Ottawa views these firms as Canadian companies as well as American -- in fact, the bailout of the Big Three included Canadian dollars -- some cross-border acquisitions have touched nerves.

Hudson's Bay, the oldest incorporated business in North America and as iconic to Canadians as is Macy's to Americans, ceased being a Canadian-owned firm in 2006. South Carolina financier Jerry Zucker bought the storied firm, a move that still resonates with a people economist Zagrus Madjd-Sadjadi says are infused with "economic patriotism."

Mr. Madjd-Sadjadi, an American academic and expert on the Canadian economy, directs the Center for Economic Analysis at Winston Salem State University in North Carolina. He sees Canada rising economically even as the United States struggles, driven in part by a booming shale oil business in the nation's west as well as a banking system that has largely been spared the trauma that put U.S. banks on the brink of insolvency.

"Canada's banks are well capitalized. Canada entered the recession later than everyone else and is coming out of it the same as everybody else," he said.

With a budget deficit equivalent to $47 billion U.S. dollars, Canada, too, faces fewer hurdles and threats to its currency, commonly called the "Loonie," in honor of the aquatic bird stamped on its dollar coin. The Canadians have run a budget surplus for the past 10 years and its current deficit is unlikely to grow exponentially as is the case in the United States.

"They don't need to put in a new health care system. They've already got one," said Mr. Madjd-Sadjadi.

They also have a trade surplus. With us. But its precise dimensions reflect yet another facet of the asymmetrical nature of U.S.-Canada bookkeeping.

In latest figures, the United States said its trade deficit with Canada surged by 41.7 percent, or $2.2 billion.

Yet Canada reported that its trade surplus with the United States narrowed to $1.9 billion.

"The U.S. is saying we have a larger trade deficit, but Canada's saying they have a smaller trade surplus. Both these things can't be right," said Mr. Madjd-Sadjadi.

What is afoot here is that Canada also has a Value Added Tax on goods, but when they are exported, the manufacturers seek a rebate. This means that a precise value is placed on the goods exported from Canada while such records are not nearly as precise because U.S. exporters don't face the same sort of need for precise valuations.

Too, much of the U.S.-Canada trade is goods, services and capital moved from a U.S. branch of a business to the Canadian branch of the same company, or vice versa.

What is clearer is that one nettlesome trade issue, a U.S. tariff on Canadian softwood exports, has been cleared up for now. Persuaded that the Canadian government was undercharging for rights to harvest timber on public lands -- what American manufacturers saw as a de facto subsidy -- the United States imposed stiff tariffs. The dispute went to the World Trade Organization earlier this decade, the United States lost at almost every step, and ultimately settled by refunding 80 percent of the tariffs collected.

No sooner had the softwood brouhaha calmed than the United States and Canada were at it again, this time over intellectual property rights.

Canada's copyright laws differ somewhat from America's, and Canadian courts have yet to settle the issue of whether music downloading is, in fact, property theft.

In May, U.S. Trade Representative Ron Kirk added Canada to his Priority Watch List for intellectual property piracy.

"We urge Canada to enact legislation in the near term to strengthen its copyright laws and implement these treaties," Mr. Kirk wrote.

The sudden hard line on Canada and copyright shows no signs of abating.

"It is important that all countries, even our closest allies and neighbors such as Canada, protect and enforce intellectual property rights in the context of a rules-based trading system," said Nefeterius McPherson, a spokeswoman for the Office of the Trade Representative.

She called on Canada to "take prompt action to address these long-standing concerns."

Precisely how to iron out U.S. companies' gripes that too much software and music piracy is taking place to our north might have to wait.

Currently, Mr. Harper's government consists of a coalition with the ever-fickle Bloc Quebecois, which represents much of French-speaking Canada. His coalition government is expected to fall before year's end, triggering a nationwide election.

Should Mr. Harper fail to win a majority, but again be pushed into a minority coalition government, big shifts on anything Canadian -- save continuing to build up that trade surplus with the elephant to the south -- will probably wait.