The US-Canada trade feud underscores the risks faced by Canadian firms doing business with the US, especially carmakers and industrial suppliers. It also points to the Trump administration's fierce determination to revamp the rules that have governed global trade since World War Two.

On the surface, this fight is over specific tariffs. Canada is objecting to new US tariffs on steel and aluminum imports. The US now says those duties were in response to Canada's high tariffs on dairy products.

The two leaders traded barbs after the conclusion of a G-7 summit in Quebec. President Trump called Canadian prime minister Justin Trudeau "very dishonest" and "weak." Trudeau's response: "Canadians are polite, we're reasonable, but we also will not be pushed around." He reaffirmed his intention to impose retaliatory tariffs on July 1, and said that US metals tariffs were "kind of insulting".

In reality, a degradation of trade terms could be a major tax on companies on both sides of the border, especially automakers and their suppliers who operate complex supply chains that often include multiple trips across Lakes Michigan and Erie.

Each country is the biggest export market for the other. Although the US has slightly lower average import tariffs, both countries operate relatively open markets, with average tariffs under 5%, according to the World Trade Organization. They benefit from advantageous trade terms with each other, thanks to the North America Free Trade Agreement, which is now being renegotiated.

However, it will be difficult for Canadian companies to avoid this fact: The US is a bigger market, so they have more to lose.

Top Canadian exports to US, first four months of 2018 (change)

· Oil, gas $26.9 billion (+9.9%)

· Cars, trucks, parts $18.3 billion (-3.8%)

· Electrical equipment $7.4 billion (+11%)

· Plastics $3.9 billion (+8.6%)

· Wood $3.4 billion (-3.2%)

· Aluminum $2.8 billion (+0.1%)

· Electric machinery $2.5 billion (+4.6%)

· Iron and steel $2 billion (+20.3%)

· Paper and paperboard $2 billion (+7.3%)

The biggest exporter of goods to Canada is the US auto sector, which stands to lose from the duties on imports of steel and aluminum from Canada. Companies aren't just shipping fully-made cars and trucks. They're also exporting parts in cross-border supply chains.

Top US exports to Canada, first four months of 2018 (change)

· Cars, trucks, parts $18.4 billion (+10.7%)

· Electrical equipment $14.7 billion (+8.7%)

· Electric machinery $8.6 billion (+8.1%)

· Oil, gas $8.4 billion (+40.5%)

· Plastics $4.5 billion (+6.8%)

· Aircraft, parts $3.1 billion (+30%)

· Optical, medical equipment $2.9 billion (+8.6%)

· Articles of iron, Steel $2.1 billion (+11%)

· Chemicals $1.8 billion (+2%)

· Iron, steel $1.7 billion (+6%)

By comparison, the dairy market that Trump has been complaining about is small. Canada considers its dairy farmers sacrosanct, and protects them with quotas and tariffs. Despite the size of the sector, little goes in or out. In 2017, it exported only $440.1 million of dairy products, with $205.9 million, or 47%, of that going to the US. It imported $618.7 million, $324.9 million, or 53%, of that from the US.

That's peanuts compared to the US-Canadian auto manufacturing supply chain which grew out of NAFTA and the hundreds of steel mills, factories and parts manufacturers that clustered around Chicago and Detroit, and just north of the border, in the 1950s and 1960s.

Total trade, cars, trucks and parts US-Canada

· 2006: $107.7 billion

· 2007: $110.4 billion

· 2008: 93.1 billion

· 2009: $63 billion

· 2010: $88.4 billion

· 2011: $97.7 billion

· 2012: $107.9 billion

· 2013: $107.7 billion

· 2014: $107.6 billion

· 2015: $103.5 billion

· 2016: $106.4 billion

· 2017: $107.3 billion

The Trump administration, however, is not wrong about its assertion that the US has been running chunky trade deficits with Canada.

US trade deficit with Canada

· 2012: $31.6 billion

· 2013: $31.7 billion

· 2014: $36.5 billion

· 2015: $15.4 billion

· 2016: $10.9 billion

· 2017: $17.5 billion

But this spat is not just about Canada. This beef is emblematic of a wider challenge the Trump administration wants to bring to system of rules that have governed trade since World War Two.

As Trump tweeted on Sunday, "sorry, we cannot let our friends, or enemies, take advantage of us on Trade anymore. We must put the American worker first!" At a news conference on Saturday, he called the US "the piggy bank that everybody is robbing."

That language signals that Trump wants to reverse the bargain made by generation of economically-liberal leaders in the US and Europe before him. They gradually reduced tariffs and trade barriers, ushering in a new era of globalization.

Because the biggest manufacturing companies were European and American, it made sense for them to set low import tariffs while negotiating market access for their firms. That way, companies like Nike, Wal-Mart and Levi Strauss could build factories in Asia, make their goods cheaply, then ship and sell them in the US and Europe at high profit margins.

That bargain is a big reason the US has such low tariff rates, and, by a long shot, the world's biggest trade deficit.

Biggest trade deficits, 2017

· US $796.2 billion

· UK $203.1 billion

· India $149.8 billion

· France $89 billion

· Turkey $77.1 billion

· Hong Kong $39.6 billion

· Egypt $33.3 billion

· Spain $30.3 billion

· Philippines $29.6 billion

· Greece $24.1 billion

The relative size of the US trade deficit is striking, even if you factor in an estimated $800 billion in service exports. (Service imports amounted to around $600 billion in 2017.)

Whatever the economic argument, companies and governments in Canada and elsewhere have to reckon with view popular in American public opinion that the deficit has gotten out of hand -- making tariffs a winning issue for Trump.