My paper "Cycles in War" addresses this question, too. I was interested in characterizing the kinds of cycles that can occur. In other words, what does the structure of a cycle in War actually look like? I simplified the problem by assuming that wars are not possible (i.e., the cards have a strict ranking from 1 to n, where n is the number of cards in the deck). Even in this simpler version I found it difficult to characterize all of the cycles. However, I did find some interesting cycle structures. For example, in the following deal of a 52-card deck cycles, assuming case that the winning card goes to the bottom of the winning player's deck before the losing card, I was able to find a way to construct a deal of an $n$-card deck that cycles, for any $n$ that is not a power of 2 or three times a power of 2.
For example, the following deal of a 52-card deck cycles.
26 46 1 7 8 27 9 28 29 47 2 10 11 30 12 31 32 48 3 13 14 33 15 34 35 49 16 36 17 37 38 50 4 18 19 39 20 40 41 51 5 21 22 42 23 43 44 52 6 24 25 45
It takes over 30,000 battles for the deck to return to this ordering. The mathematical argument for why this deal cycles is in the paper
, which has been accepted for publication by the journal Integers but has not appeared in print yet. Among other things, the re-loading rules do make a difference, as in this example and as other people have already noted here. Also, given that characterizing cycle structures when wars are not possible turns out to be difficult (or, at least, I found it so), one should expect that characterizing cycles in the standard version of the game in which wars are possible would be even more difficult.
Edit: The paper has now been published on the Integers web site, in the games section, as Vol. 10, Article G2, 2010, pp. 747-764.