Someone asked me recently if there was any advantage to being able to put your stick down second for a face-off (as the home team's centerman does.) Let's look at the home-road numbers for 2003-08:
There is a clear home-ice faceoff advantage with a very small portion of that advantage due to regular centermen lining up against wingers or inexperienced players who otherwise took fewer than 100 faceoffs in 5 years. In addition, 86% of players who took more than 100 faceoffs had a home-ice advantage.
Obviously, the home-ice advantage is not necessarily due to putting the stick down last. Not that I've actually analyzed this, but because the home team also has the last line change, this may result in the home team rarely having a poor faceoff-taker lined up against a good one in critical situations.