In the "old days" I sometimes composed my photos diagonally, but I've not had success doing that in the "modern" era, because there's not really any nice way to present it digitally.
Is there a reason to take diagonal photos? Does the composition have particular strengths?
I still compose diagonally on a regular basis when shooting bands, I find this maximises what I can get into the frame, and the resulting images work both mounted diagonally and in a regular upright orientation:
I agree that presenting other images like this wouldn't work, for example if you do a diagonal composition of a shot with a horizon it will just look wrong. However a lot of digital photographers still regularly print their photos, maybe not as often as they should. Perhaps if more people adopted this form of composition people would be more inclined to print.
Maybe this is not entirely related, but abstract shots fit quite well with diagonals... :o)
As I said in a comment, I don't think it has ever worked for me (viewing these types of photos).
All it has ever done has made the viewer tilt there head to see what's really happening in the scene.
Portraits in particular may be taken at a moderate angle as it may serve to balance the height of two different subjects or to capture the off angle of the subject (they may be leaning or something).
In my own band photography as an example, I rarely put a "roll" on my photos as it puts me off whenever I've seen it elsewhere.
Take the example above and compare it to a shot where no tilt is (photo quality aside):
Now as an example (of course, my opinion), here is a portrait I've taken where a moderate tilt has helped to frame the subjects where the environment is only a secondary part of the photograph.
I think it really has to do with the subject, not just throwing a 45 degree tilt in for the hell of it. Take this shot as example. I think this personally works great as the subject (guitarist) is framed vertically even though he is in fact leaning backward.
So in my opinion, large tilts never work unless they are done to keep the subject framed properly.
More: I think it actually also goes against the natural tendency for humans to remain upright for the most part of our lives. Do I go to a concert (keeping with the band photography) and look at the band with my head tilted?
I recently took a picture where I instinctively tilted the frame slightly while composing:
Just to see, I straightened it out in an editing program, like so:
In looking at them both for a while, I prefer the first one. There's a greater dynamic sense, and it has a more casual aesthetic — but the parallel line of her arm and the right edge of the image provide a stop and bit of static balance. The leveled image is too much "okay, I'm standing here".
This isn't a 45° angle that would make the image a composition about diagonals, but I think it illustrates the value of non-squared lines as a element in an overall composition.
Other than the issue of presentation, there's another thing to consider in the digital age. If you want to print a slanted film photograph turned to be straight (or vice versa), the only loss is the cropped corners/edges. If you rotate a digital file by an arbitrary angle, pixels have to be re-interpolated, which is inherently a very lossy operation. It's like running a blur filter over the whole thing.
In my example above, notice how the sparkle has gone out of Anya's eye. I didn't treat these two images differently in any way except for the rotation. After that, they're both scaled down with an identical filter. I did the work really quickly and from a JPEG; I'm sure more careful work could avoid that particular damage, but there is a very concrete example of the loss I'm talking about.
That doesn't break shooting in this way, but makes it a bigger decision. Of course, one can always keep the digital file as-is and turn and crop after printing, but that different from the workflow many people who shoot digitally are accustomed to.
In response to Matt Grum's comment below: the test of rotating and then rotating back by the exact same amount doesn't show the whole problem, in that the operation causes blur, but that blur is largely reversible if you do the exact same thing in reverse. If you just leave the image rotated and continue to do other things (like, say, print), the loss is made permanent.
I tend to find that I'd rather shoot a picture 'straight' most of the time and introduce a rotation after the fact in post production, because that way I give myself the option. Of course I love spending time in post-production, so I do understand that my answer isn't optimal for a photographer that wants to do everything in camera in order to minimize time spent in post.
I think there are certain types of shots that lend themselves well to rotation... shots that are dynamic in nature and/or are intended to convey a sense of movement. I also see a lot of '"rotation for rotation's sake" out of some photographers who don't necessarily have a great eye for composition, but think that it makes their shots 'better,' because photographs with severely dutched horizons has been a bit of a 'fad' in the advertising industry over the past few years...
Personally I think that dutched composition is a tool in the toolbox to be used sparingly... Like a fisheye lens. The occasional picture shot with a 'specialty tool' can add life to a set, but if every picture is rotated it gets cliche real quick.
I have experimented in adding a frame within a couple of my photos to achieve just such a thing: