While looking for more examples of this process, I stumbled upon the answer.
Why is Harry Reid always voting against his own plans?
Forty U.S. senators voted to block a final vote on Chuck Hagel's nomination to be defense secretary. Of those, 39 were Republicans opposed to the nomination, at least for the moment. The other was Harry Reid. It wasn't that Reid opposed Hagel — far from it. ...
So what gives? In articles like the one I wrote on the Hagel filibuster, the short explanation we give is that Reid voted no "for procedural reasons" or because a no vote "allows him to bring another cloture vote in the future." ...
As Sarah Binder, a Senate rules expert at George Washington University, told me, it's not that the majority leader has to vote no. It's that somebody on the winning side of the cloture vote — in this case, the side voting against cloture — has to file a "motion to reconsider" if the matter is to be taken up again. "I suppose the broader parliamentary principle here is that it would be somewhat unfair to give someone on the losing side of a question a second bite at the apple," Binder explains. So the rules provide for senators whose opinion has changed to motion for another vote, whereas those whose opinion stays the same don't get to keep filing to reconsider.
Reid, and other majority leaders before him, have developed a clever workaround: Just change your vote at the last minute if it looks as though you're going to lose, then move to reconsider. In theory, any supporter of the bill or nomination in question could do the same, but traditionally it's been the majority leader.
Senate rules say that after a failed cloture vote (ie. the bill did not get the 60 votes needed to overrule a filibuster), a "Motion to Reconsider" (that is, a second attempt at cloture) can only be called by a member of the winning side of the vote.