one scene that I've never been able to understand

I always interpreted this scene as triplette taking a look at the girl and somehow figuring out that she has a loose character. your interpretation is valid as well.

I always saw that scene as a woman trying to take her thoughts elsewhre and remove herself from her current emotional state. They were fighting and then comes the knock on the door. They were celebrities and were looking out for their reputations. I'm sure they didn't want it to leak to the press that Bill, Mary and Tom were having internal problems and were maybe breaking up. Of course, that's exactly what was happening but... no need in everybody knowin' Gotta protect yourself.

I think in a more tacky version of the movie those 'odd looks' as you call them would be intercut with shots of her cleavage. He was just checking her out, I don't think there's much more to it than that.

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http://www.youtube.com/watch?v=JIMBC7PA_uo

My interpretation is that this is a way for Mary to show her disdain for both Triplette and Bill. First, she says that they are Democrats, and wouldn't appear in favor of the Hal Philip Walker. Second, she decides to show disrespect by buttering up her face. Where I do agree is that Triplette is probably always on the make, and eyes up Mary and in the last scene before the assasination, he also eyes the other ladies on the line on stage.

I kind of focused on Triplette in my last viewing; he does that look a lot, and at other women, including LA Joan.

Datta, dayadhvam, damyata.

Interesting question. Here's how I always read it. Triplette is a political advisor. Like a salesman or a politician, he reads situations immediately and adopts an attitude and a persona that will get him what he wants. Pure opportunist, pure chameleon: at various points in the film he presents himself as progressive, as a good ol' boy, as a bully, as sympathetic...whatever it takes. His response to Mary and Bill is initially puzzled (there's clearly more going here than meets the eye), then an increasingly shrewd, amused and curious attention paid to Mary as he realizes that in fact SHE will make the decision.

This is Michael Murphy talking about the scene on the new Criterion blu-ray:

"I was standing with Bob, watching the rehearsal, and he had his hand on my shoulder to give me a cue. Now I'm watching this thing, and I leaned over and whispered in his ear, 'Bob, I should play this like I've got the hots for her!' Because [my character] was such a straight guy, and to see this kind of hippy-dippy weird chick, you know? And I whispered [that] to Bob, and he said, 'Just do it!' and pushed me onto the set."

I don't know how any hetero male could keep from looking at Claudia Raines.

Nobody's looking for a puppeteer in today's wintry economic climate.