I often re-read books that I've enjoyed, and I don't think it's a strange thing to do at all. With poetry nobody would think it odd to read it several times over, so I don't see that it should be different with fiction, after all, reading is a pleasure!
I don't have any set pattern for re-reading LotR, I often just pick it up to look something up and find myself a few hours later having read far more than I intended to, I just get swept up in it. This goes for anything by or about Tolkien too.
How strange though that there are so many other Tolkien readers who also like Austen - I also re-read Austen. What does this mean? I suppose that we like good writing!

What else do I re-read? I've read most of Thomas Hardy's novels several times over - especially
Jude the Obscure, and I also re-read anything by the Brontes over and over. Other favourties I read many times are
Possession, The Secret History, HDM, Gormenghast, Dracula, The House of the Spirits, and in the non-fiction genre, anything by Bill Bryson. Sometimes I just read favourite 'bits' over, sometimes the whole thing. I could make a huge list here actually, because once I've read a book I invariably go back for more. I will even read favourite books from childhood, and never get rid of books because if they were good enough to buy they are good enough to keep!