Which sounds better:
When Canadians do initiate conversations, they tend to be more reserved than Americans do.
When Canadians do initiate conversations, they tend to be more reserved than do Americans.
Okay. I'm going to remove the first do in each statement, and come back to it later.
Both read fine, but which is stronger? There's an interesting thing happening in the second, which we can see if we rewrite them to not use do-support:
We can't actually do that with the other one and come out with the same meaning, the closest I can get is:
Which is more than a simple, direct rewrite. Indeed, it's quite different, indeed.
The interesting thing about the second of the first sentences, is that we have the verb before the subject. Much of the time, this plain won't work in English. Some of the time, it'll be understandable, but sound pretty weird, or match some archaic constructions.
Here though it's a mild, but very pleasing example of hyperbaton, and it leads to the sentence ending on one of the groups we are interested in, rather than the relatively weak do. (The reason it's mild, is that verb-subject is nowhere near as rare with do, be, have and a few other verbs as it is with most, so it's arguably not really hyperbaton at all).
Now, let's consider StoneyB's suggestion from the comments, but again with my edit:
Okay, here rather than using do to match the earlier verb phrase, we use are to match the to be in that earlier phrase, and hence evoke it.
And it pretty much matches, as far as that mild hyperbaton goes. Again, I prefer the second.
I still prefer the do form to the are form. Partly, at least, this is because it could be seen as matching how reserved Canadians tend to be in a given situations, with how reserved Americans are generally. It's one of those mild ambiguities where I wouldn't actually misinterpret it that way, but it does weaken the impact. (We could also leave out the are and have "...tend to be more reserved than Americans", but that has the same issue even more strongly).
Okay, now to go back to having the first do in there:
When I first read the two sentences, they seemed wrong to me, and the reason was that I was matching the do at the end to the do near the beginning. Now, that was a misreading, and when I read them again more carefully, I realised the meaning. Still, even now it reads to me like that repetition—even of such a weak word as do can be—is deliberate and meaningful, and I trip over it mentally. (Incidentally, I found that effect even worse with the other sentence suggested).
Now, presumably that do is there for a good reason. At a guess, an earlier statement is saying that Canadians don't often initiate conversations, with this sentence addressing the exception to that. It's a good reason to have do in there; the last thing you want is the reader to react with "why are you talking about Canadians initiating conversations, you just said they don't!"
But I do still find that repetition to be a bit of a burr.
So, if you can remove that do without causing problems in the wider context, I would recommend:
If however that seems contradictory in the context, I'd recommend:
Again though, this is offering an opinion of effect, which is subjective.
The more general answer to the more general question, "Should one invert syntax for the verb “do” in a comparison?" is that you can, but you don't have to, and while I may favour it in this case, others may disagree, and I may disagree myself with another example. "So baby talk to me, Like do lovers" just wouldn't work as well as "So baby talk to me, Like lovers do".
I am hesitant to write this because I am new here and others on this site are far far more knowledgeable than I. That said, and assuming the first do is for contrasting with a previous statement, I would go for
The meaning here is that when Canadians, who do not often do so, initiate a conversation they are more reserved than Americans in the same situation. I do not understand the need for a final verb. Like the Canadians, the fictitious Americans in question are also initiating a conversation. They are not doing and they aren't being anything. Both the final are and the final do seem unnecessary.
Granted that this form makes the sentence slightly more ambiguous. It could now be taken to mean that Canadians, when initiating a conversation, are more reserved than Americans in general as opposed to Americans initiating a conversation. Nevertheless, I feel that this form is clearer, easier to read and more natural.
I'll delete this if anyone comes up with a better answer, but I think both versions are grammatically fine. You could replace do with are, but I see no reason that should be required.
As to which sounds better?, I prefer the second. It sounds more measured and formal, which is in keeping with the rest of the sentence. Sticking the auxiliary verb do (or are) at the end sounds more like a spoken form where the speaker hadn't quite thought out how he was going to finish until he got there (and arguably would have been better off not saying the last word at all).
EDIT: As has been pointed out, you don't need to use "do" at all. It would be pedantic to suppose the sentence was defective because of any potential ambiguity (i.e. - that Canadians when they happen to be initiating conversations are more reserved than Americans in general). But that's not what the question asks.
I also acknowledge Jon's point that the reason OP's second version is "better" is that Americans is the operative word in the clause, and as such has more impact in the final position.