There are some people who believe that checkerboard tile floors should be laid on the diagonal. For others, straight up and down is perfectly fine. I suppose it's an aesthetic preference, though I'm sure one style is more classically and architecturally correct than the other.
If my memory serves me correctly, and lately it hasn't been, Dorothy Draper had a very definite opinion on the matter. I believe that she was an advocate of the diagonal method, though her projects show that she flirted with both styles. Nevertheless, after finding numerous images of black and white floors, I think I prefer tiles laid on the bias. Visually, diagonal tiles draw one's eye into the room. And there's something quite elegant about it too. The up and down fashion, like that seen above in this movie still, truly does look like a checkerboard, but to me it looks a little blocky. (The image at top also is a great example of why larger tiles look better than small ones.)
When I was trying to find images for this post, I mostly found floors with the tiles laid diagonally, which leads me to believe that most designers prefer this style. Which do you like?
Hampshire House was one of Draper's earlier projects, so perhaps she had not yet formulated her opinion about diagonal tiles.
Dorothy chose large scaled tiles laid on the diagonal for the floor at Quitandinha.
Mrs. Kersey Coates Reed House, David Adler architect
Remember this home decorated by Celerie Kemble? The tile is actually a painted wooden floor, but the faux tiles were painted on the bias.
Woodson Taulbee's home in Old San Juan, decorated by Billy Baldwin.
(Image at top: Movie still from 1929 movie "The Show of Shows"; from Screen Deco (Architecture and Film, 3.). Image 2 and 3 from In the Pink: Dorothy Draper--America's Most Fabulous Decorator. Photo 4, David Adler, Architect: The Elements of Style. Image 5 courtesy of Celerie Kemble: To Your Taste: Creating Modern Rooms with a Traditional Twist. #6, Billy Baldwin Decorates: A book of practical decorating ideas)