I've never yet read LotR in any kind of 'unusual' order. I read it straight through the first time around, at the time I did jump back and look at bits over again, just to clarify the story, though I wouldn't call it re-reading as such. I also started with The Hobbit - as I was 'borrowing' the books, my brother told me that I couldn't read them
at all if I didn't start with The Hobbit.
Nowadays I will have a complete read through now and then, though I've not done that in a while due to reading it through in any case, albeit slowly, for the CbC discussions. But I always go looking for something, just for a passage, and end up reading chunks I hadn't meant to read. Sometimes I can find myself a few chapters on and have to stop myself.
I've never had any urge to read it in the wrong order just to see if it is any different because I think "if it aint broke, why fix it?".