This is an extension of something I wrote elsewhere, but: I've always thought that one of the reasons the mirror routine in Duck Soup is such a great version of that routine (which had been done many times before, and would be done many times after, but -- at least on film -- never as well) is that both Groucho and Harpo know exactly what is going on.
The usual way to do that routine is to have the person in disguise try, with some success, to fool the other person. That makes it a clear straight man/comic routine: one character is in on the joke, and the other isn't. But in Duck Soup, Groucho never acts like he's fooled by Harpo. He knows he's not looking in a mirror. He knows this is a guy made up to look like him. He could just reach out and grab Harpo (the way he finally grabs Chico at the end of the routine). But he's intrigued by how successful Harpo is in imitating his movements, and he's determined to make Harpo slip up.
So the routine as it plays out is partly an exercise in timing and simultaneous movement, but only partly. A lot of it is just a battle of wits between two characters who each know what is going on. Sometimes Groucho seems to have the upper hand, but Harpo turns the tables on him. Sometimes Harpo pulls off a feat that is physically impossible (producing the right object out of nowhere when he clearly had a different object behind his back), and sometimes he wins in a very realistic way (standing still while Groucho spins around).
And as the routine goes on, Groucho becomes so obsessed with the game that he loses sight of the objective: Harpo drops his hat and Groucho hands it back to him, then Harpo goes totally out of synch with Groucho, but Groucho then starts to think of some other way to make Harpo drop the act, not realizing (or not caring) that it's already happened. Groucho has forgotten why he's doing all this; he started the scene with a goal, and now he just has a ritual that he's going to keep repeating obsessively. If Chico didn't wander into the scene, making it impossible for Groucho and Harpo to keep up the pretense, they'd conceivably keep on doing it forever.
I have no doubt that this is the way this routine was sometimes done in the better Vaudeville and stage versions. It may even have been done this way on film before. But this is not the way it's usually done; it's usually just done to show us that two performers can move in exact synchronization. And that's just one joke, really. It may be a well-executed joke if the performers are good enough, but it's not as funny as Duck Soup, because it doesn't go through all those different stages and have as much back-and-forth between the characters.
That's why the version of this routine from I Love Lucy never did it for me. It's well-done in the technical sense, but the way it's written, Harpo is basically acting like a moron for a large portion of the scene. He acts like maybe that is his reflection, and he's just testing to see whether this is so. Groucho doesn't have to play dumb, because in their version of the scene, he's not taken in. (And this even though Harpo-as-Groucho looks a lot more like his brother than Lucille Ball looks like Harpo.) In Duck Soup it's a scene about two smart people trying to score points off each other, instead of a scene about an idiot being fooled by what is obviously someone in a costume.
Abbott and Costello's "Who's On First?," as I also said in that other post, has an element of this as well. It's pretty clear as the routine goes on that Costello has figured it out. That's the point of him saying "third base!" along with Abbott: Costello knows that I Don't Know is the third baseman. But he can't just cut it short and phrase his question in a different way. You can interpret it as Costello being trapped by the rules of this comedy routine, or Costello being stubborn (like Groucho in the mirror routine) and refusing to give up until he gets Abbott to give him the answer he wants in the exact way he wants. But there is no sense, the way they do it, that Costello is an idiot who can't figure out that "Who" is the name of the first baseman. The dynamic is that of two guys who will not give up phrasing things in exactly the way they want: the other one will have to change his wording first.
Update In comments, Griff argues that I'm being too hard on the Harpo/Lucy version:
I don't believe that Harpo is fooled for a minute, though he may mime some amazement to the audience; he's determined to play this through so long as Lucy is. It's a sketch compared to the full-blown sequence of the McCarey picture, but both Harpo and Lucy do play it seriously. No small amount of the scene's comedy comes from Lucy's desperate hopes to fool Harpo, and his obliging, energetic, even generous following along with her.
That well may be the case. The way I always interpreted the scene, based on Harpo's amazement and confusion, is that he is fooled or at least that he doesn't fully know what's going on. But when I looked at the scene again, I realized I'd forgotten an important detail (not having seen the episode in a long while): he does see her duck out of sight at the beginning of the scene. And that's presumably meant to establish that he knows at least part of what's going on (someone is impersonating him). So it may be more accurate to say that he is humoring this mysterious Harpo imitator for as long as she can keep it up. I still think the way the scene plays out, and many of Harpo's gestures, indicate that he doesn't really know what's going on -- like his shrug and the way he feels his face as if to check whether he really looks like that younger person in the mirror. But it's not as simple as him just being clueless all the way through.
Here is the scene, anyway: