International Chimeny is a classic route that shouldn't be missed. Start on the west side of Spire Three. Pitch 1 climbs up an easy chimney (class 4, easy class 5) between Obelisk and Spire Three. Continue this until at the base of the International Chimney. Pitch 2 climbs the obvious line. This isn't a chimney per se, in fact, no chimney moves are encountered. It is more of a detached flake from the main massive. Climb the large crystals to a belay at the top of the flake. From here, I believe that the original route must traverse further right, but most people go straight up the insipient crack/groove to the summit. This pitch is probably 5.8 and deserves an s (serious) because of the lack of pro and a nasty possible fall into the chimney. Climb with care to the easy, slabby finish.
Descend by doing a two-rope rappel to the ground.
Protection
Bring a standard rack. The first two pitches have good protection. The 3rd pitch is very thin for gear. Aliens might be useful?
I agree that the top is 5.8 not 5.6. With double ropes, some protection can be set up across the "chimney" from the incipient groove that leads to the top. This protection is better than nothing, as it will prevent a horrible slide down the chimney.It's a great climb, and a fabulous view on a clear day.
When I climbed this route, I did the traditional 5.6 second pitch that starts from the southern end of the corridor. Though it looks unlikely, it is possible to get in some small gear and then throw yourself into the squeeze chimney. Ten to fifteen feet of grunting later and you'll find and easy scramble to the summit. I got a good chuckle when reading the new Extreme Angles guidebook to the Needles because they rate that move into the squeeze chimney 5.9! Though the leap was a bit freaky and committing, I didn't think it was anywhere near that hard.