Actually, I don't think that's true. Here's the rule, as stated in the first post in this thread:
"Popular phrases, Such as...can go before or after previous word." In all of the examples given in the first post, the second word of a phrase was the first word of the next phrase, but the rule is sufficiently ambiguous that it doesn't appear to require it. When mmstan started this thread, I asked him what the difference was between his word game and "Two Words with a Twist," the word game I had started a little while earlier (in which one can use either word from a previous phrase as either word in the next phrase), and he acknowledged that there wasn't a difference.
There was, when I started "Two Words with a Twist," another word game in which the second word of a phrase had to be the first word of the next phrase, but I can't remember the name of it and I think people just stopped participating in that thread. If someone wants to find that thread, or start a new one with that rule in it, that would be cool. The "second word of a phrase must be the first word of the next phrase" rule makes the game a little more challenging.