That is what a Camera Finder is for. There are 7 current DSLRs with a rotating display. Those do exactly what you ask for twisting away from the camera body rather than simply tilting up and down as some articulated displays do.

For a specific recommendation, the truth is that they are all good, particularly if you cannot tell the difference. Newer ones tend to perform better so something like the Nikon D5100 would have one of the best image quality for an entry-level model. There is also the Canon 60D on the list which is more advanced model that is faster to operate and better build quality. Keep in mind that it is extremely important that you save some of your budget for a good lens, otherwise the performance of your new DSLR will be limited.

NOTE: I see someone gave you an partly incorrect answer in the comments. That list includes 4 cameras that are not DSLRs (the last additional one is a discontinued model). Those are SLDs which have interchangeable lenses like DSLRs but not an optical (reflex) viewfinder.