If you guys participated in burning the White House as you said, then you also were responsible for dropping the Atomic Bomb on Japan. Just following your logic.
It's not quite like that. What would be more like what happened would be if a Canadian bomber had been order by the US to drop A-Bomb on Japan.
In 1812, Canada was legally a colony of Britain, but settlers had been there as long as Americans had been settling their part of the continent... namely, hundreds of years, such that the colonialists had basically cut their roots from Europe for centuries.
However, the *administration* of government was still being driven out of London, so Sherman marched north, figuring he'd be welcomed with open arms by colonialists being liberated from British control, which was not an altogether unreasonable presumption, except for a few historical issues that were somehow overlooked.
Specifically, when the American Revolution happened, what's not talked about a lot is that fully one third of American colonialists did not support it. What's also not talked about much is why not all of the colonies joined in, because there were more than 13 colonies. To the north there was Nova Scotia, Quebec, and Newfoundland.
Newfoundland never joined the revolution because they were so remote they might not have heard about it, plus they were very comfortable doing what they did, which was catch and sell fish to the Brits. Quebec would have joined the revolution in the blink of an eye if there hadn't been a language barrier, and I don't know why Nova Scotia never joined in; probably for the same reason as Newfoundland, namely, they were comfortable being fishermen selling their catch to England.
When it became clear that the revolution was going to succeed, there was a point where Boston harbor had as many as 300 ships anchored and waiting to take wealthy loyalists who could afford the passage a trip back to England.
But a lot of them could not afford the trip back, and even if they could, they wouldn't have had any land to go back to, so those ones packed up and headed north.
Some of them went east into Nova Scotia, which was larger then. The areas that the Loyalists settled into eventually split off from Nova Scotia to form New Brunswick.
Some went more towards the center, into Quebec, which was much larger then. Those Empire Loyalists settled in-and-around the Niagra peninsula, and eventually resulted in creating Ontario, split off from what had been Quebec territory.
When Sherman marched north, he forgot that a lot of the people he was heading forth to liberate had explicitly moved there to *not* be part of the United States, so when the British governing authority put out a call to arms for recruits to defend the invasion, they found no shortage of volunteers.
In terms of lines of authority it was indeed a British command, and Canadians will insist on it being noted that the grunts actually doing the fighting were Canadian, but if you *really* want to dig into it, what Sherman was fighting included a lot of American expats who'd been happy with the status quo and who hadn't wanted to see the Revolution succeed... so they had a bit of a chip on their shoulders. It might have been the first time America marched forth to liberate a people, presuming they'd be welcome, only to be surprised to find that they were not welcome.
Jumping forward to 1867... the issue was that British North America had been watching with concern the American Civil War, and were very concerned to see how it ended because, oops, did they tell you that the governing administration of British North America had been supporting the south? The *people* supported the north, but the *governing administration* had been supporting the south, and at the end of it, the Yankees found themselves with the largest standing army in the world at the time... all seasoned combat vets... which is what you have to be scared of.
So, some home-grown Canadians of the educated class freaked at what would happen if Uncle Sam were to decide to turn their seasoned, standing army around and march it north to settle an old score from 1812, so in a mad-dash they scrambled to do a couple of things, including pull most of the key executive decision making processes for managing defense over to this side of the Atlantic so they wouldn't have to wait for the 2-3 week turnaround for orders to go between BNA and London, plus change the name to something very New World like Canada and get away from being called British North America (you know, so you can say to Yankees, "Hey look, see, we're not British North Americans like those British who supported the south... we're Canadians!... we *like* the north... heck, we're true-north strong and free!"), and the first thing they did was go on a mad-panic construction of canals connecting upper and lower Canada so they could move forces back-and-forth anywhere along the line to meet Americans without having to use the Great Lakes.
But the Brits had insisted on maintaining a few strings attached, and it took a long time, over several significant steps, to finally cut those string. Crumb, I think the last legislative string wasn't cut until the time of Trudeau.
So you see, that War of 1812 wasn't as simplistically Canada versus the US as many would like to think. It was a British command, and a lot of the grunts were American expats pissed off about feeling required to leave their homes simply because they wanted to maintain loyalty to the crown.
(Imagine how it was for the French Canadians when a bunch of English speaking Empire Loyalists packing a grudge pushed into their eastern frontier to settle down to eventually call themselves Ontario, and because those future-Ontario settlers are carrying a chip on their shoulder about feeling victimized and forced into exile for having supported the crown, they take it out on the indigenous Quebecois, hammering on them to honor the British Crown, when Quebecois were happy enough to peacefully forget that the British Crown existed if everyone would just shut-up about it, but the Empire Loyalists wouldn't let bygones be bygones and wouldn't stop cramming that bad salt down their throats, leading to Quebec's attempt to split from British North America in 1837-38, etc. etc. yadda yadda... but maybe that's for another thread... I'm just saying, man, the more you learn about the history of it all, the more you'll see that Quebec separatists have a point...)
Anyway... the last time there was any sort of real border dispute between the US and Canada was in 1859 when a skirmish called the Pig War - so called because it was triggered by the shooting of a pig (which in the end was the only casualty of the war) - happened over the question of where to draw the border through the waters of the southern Gulf Islands off the west coast.
It was eventually settled by third-party arbitration in 1872, and since then there haven't been any border disputes. There were some questions about US whiskey traders moving into southern Alberta, who got chased out with creation of the RCMP, and the 1903 tribunal establishing the border with Alaska was very peaceful and orderly (I'm jealous how Alaska got that huge pan-handle).
Otherwise, there hasn't been an issue. Both sides just got used to each other being there.
So what would I do if there were an invasion?
It would mean the border's disappeared, so I'll wait for a day when the weather is good to talk to the troops and find out if any of them want to trade houses, and if any takers, I'll trade him the keys, tell him not to forget to water the dandelions, and I'll tell him to give me a call the next time he's around and we can go out for a drink and he can tell me his war stories because I'm going to take the new house and play with it on the real estate market until I get a summer home in Tillamook and a winter home in Phoenix.
By the way... totally off thread, but... guess who maintained an embassy in North Vietnam and was the relay for diplomatic communications between the US and North Vietnam during the Vietnam War?