In my first post, I showed a glitch where doors could be cut in half by switching them to your off-hand at the same time as placing them. One of the helpers said that I was using a modded version, which was true, I was using OptiFine.

So I decided to try it in vanilla anyway, and the bug was still there. The attached video should prove that it's vanilla Minecraft.

My original post:  You can make a door be cut in half by placing it at the same time you switch it to your off-hand. Not only that, but the door stays in your inventory, allowing you to place it again. To top it all off, if you open the 'cut' door, it completely disappears. The door is invisible to other players, making it look like they walk through it, but the person who placed the door can't.