I'm working on a logo design and the client is concerned because they've seen a similar type treatment. I know this is a very common approach, but I'm wondering if there's a specific name for this kind of type treatment/arrangement?

You have combined two treatments.
One is what I have always considered to be the “title page” technique of breaking down a sentence into phrases/components which are given dramatically contrasting settings.
This does indeed appear to be quite popular now in all-type posters with clever/inspirational messages.
(These need not be symmetrical.)
I call the style “title page” because I believe that is where it originated, before becoming prevalent in posters and advertisements in the early 19th century.
Although “title page” is not a generic term, “tombstone” is a category I’m familiar with; I came across it early in my career in advertising, when symmetry was frowned upon as being dull, old-fashioned and unimaginative.
The other is the escutcheon/cartouche/label, which is not a free-standing item, but generally adorns an object as a logo or label, and has a roundness which often informs the disposition of type and lettering on curved paths.
As Joshua notes, there is also the dimension of “class”, which relates to the type styles used, more than the layout.
Hipster branding http://hipsterbranding.tumblr.com/
Look at Dribbble to find hundreds of similar treatments even more similar to yours. Your client is right to resist a typical trendy style and push for something unique and on the mark for their brand DNA.
Joshua - thanks, I was aware this approach has been around for quite a while. I came across this while digging around online: http://www.flickr.com/photos/double-m2/sets/72157624084389178/
Nick - thanks, good info.
Luma - the design I've come up with is certainly not trendy or hipster, and is very appropriate for their business, a financial services firm. I am aware of dribble and the hipster trend - but what makes a design 'hipster' is more than a particular arrangement. The two samples you posted have nothing in common btw.
Chris, if you don't have this site bookmarked it's a good one to have as a 'go to' for inspiration, not only in the 'tombstone style' but also most everything else graphic old school/new skool...
http://www.graphic-exchange.com/home.html
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I apologize if my comment was too accusatory, I can see how it could be interpreted that way. Without seeing the actual work you have done, and since you were asking for the name of such a design style, my mind went right to the hipster branding site that I came across recently. It is an interesting question to ask "what makes these designs 'hipster'?" I am not exactly sure myself, but I do get a sense of the style and feel that seems trendy. The image I posted seems like it was a bit confusing too. It reads "THIS IS the neue This." An attempt to show the Helvetica trend replaced by the left hand style. I didn't make the graphic, I just found it and thought it illustrated what I was trying to say. Hopefully I was still able to be helpful and provide some food for thought at the very least!
Chris, I love that Flickr link. Some very interesting examples.
I wonder if rounded layouts could have been inspired by traditional seals? Round seals have been used since ancient times and perhaps the style you're referring, though obviously different in several ways, was partially inspired by that look?

I was thinking about the first image in this post, but on second thought it might not be that relevant to this discussion.
American Ralph Lauren designed and produced (in China lol) team U.S.A.'s apparel http://www.teamusashop.com/ ...hence the polo thing.
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Here's the blog I ran across yesterday referencing the term "badge" also "type lockup" was somewhere on the page as well. http://allanpeters.com/blog/?p=2322
There was mention of title page designs earlier on in this thread. In the event you're still digging for possible inspiration, or you're just itching for a jolly good read on the subject, I'd recommend, "The practice of typography; a treatise on title-pages, with numerous illustrations in facsimile and some observations on the early and recent printing of books (1902)" by Theodore De Vinne: http://archive.org/details/practicetypogra02vinngoog
It's a title you can choke on, but the book covers historical practises and how they became standard conventions; plus there's a turn-of-the-century plea for rational thought and a push for simplicity when typesetting those longer titles!
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