oh, I love backgammon, and am pretty good at it...
but it's not the kind of game that could have much to do with LOTR...
of course one could theoretically even use figures like chess pawns instead of the normal small rings
the normal colours for backgammon pieces are black and white, so maybe a dark forces /good forces might make sense...
you could also decorate the pieces with tengwar/cirth runes.
if you want to choose homes I suggest The Shire and Rivendell on one side, Lorien and Anduin, Isengard and Rohan and finall Gondor and Mordor
this way the road would suggest for one player the road taken by Sam and Frodo, and for the other the road taken by the Nazgul in their Hunt for the Ring until they reached the Shire
anyway...a very nice game