Some clients allow you to specify a different folder name, while other only allow you to specify where the folder is located.
For instance, say I'm downloading the Rabbit-Force Clannad AS torrent. It is currently located at '/media/storage/leeching/[Rabbit-Force]_Clannad_-After_Story-_[DVD_H264_AAC]'. If I move it to '/media/storage/anime/[Rabbit-Force]_Clannad_-After_Story-_[DVD_H264_AAC]', I can still seed the files (after notifying the client), but I cannot if I move it to '/media/storage/anime/Nagisa_Sucks_lololol'.
I'm not sure which clients support renaming (rather than just relocating).
I use symlinks to create an easily-browsable alternate directory. You can use shortcuts or
NTFS symlinks to achieve the same thing in Windows. There are probably Explorer extensions written for symlinks so you don't have to drop into the command prompt.