I don't have the book at hand right now, but if I remember correctly, Dvoretsky's endgame manual has some instructive examples.

However, this is a kind of endgame where I find it difficult to trust any general guidelines; it often comes down to concrete calculation. One general guideline I can give, however, is to always calculate whether the defending side can use the frontal defence and invite a rook exchange (in this case, with ...Rc8), either before or after checking the White king.

In your case, the horizontal defence is only used temporarily to chase White's rook away from c3 (the optimal square, where it defends the pawn while keeping Black's king cut off). If White replies to 1...Rh4 with 2.Rc1, the only move that draws is 2...Rh8 switching to the frontal defence (yes, I needed a tablebase to confirm that!  ). I find it hard to give any general explanation why 1...Rh4 loses while 1...Re8 draws, but by concrete calculation, I can see that 1...Rh4 forces White's rook to a less than optimal square, since 2.e4 Rh3+ leads to a drawn pawn endgame. And therefore it makes sense to insert that move.

I gave another example of this kind of endgame in an earlier thread:

http://www.chesspub.com/cgi-bin/yabb2/YaBB.pl?num=1243588570/15#15

In this case, I found a winning line by cutting the king off horizontally instead of vertically. But again, I cannot tell you by any general guidelines why 1.Rf7 won in that case while the apparently more natural 1.Re5 would have drawn. I had to use concrete calculation.