Why are my handlebars difficult to turn in one direction but not the other?

There are a couple possible causes. Proceed with caution.

• One cause of this is if a ball bearing, or a broken piece of one, has gotten out of place and is now grinding between the steerer and the headset cup or frame. This could happen either due to extreme wear or someone opening up the headset and not being careful enough when putting it back together. This can wear a groove in the steerer that can lead to failure, and it's also imaginable that it could lock up the fork's turning while riding in a dangerous way.
• It's possible that simple wear could wind up causing this, but only if someone then adjusted the headset with the goal of taking out all the play, and then it caused it to stiffen up elsewhere. If you slowly acquired this problem without anyone putting their hands on the headset, simple wear probably isn't what happening, although extreme wear (balls and retainers tearing themselves apart, etc) could cause it.
• A bent steerer causes this. So if the bike's been crashed or smashed into something, that's important to look at.
• Poor headtube facing causes this, although typically you wouldn't experience that as an acquired problem. I could imagine a situation where acquired headtube ovalization causes it too.
• Incomplete cup insertion always causes this, but again that's not an acquired problem.
• If the steerer is bulging, as in an overtightened quill stem wedge, or has any other irregularities that could affect the alignment of the headset, that can cause it.
• It's uncommon, but threadless stems with poor facing (uneven surfaces) can cause this.