Can I rotate patterns?

Unfortunately, there's no way to rotate patterns. Unlike furniture, which will drop in a certain direction based on where you're facing, patterns do not behave this way. They are always placed in a fixed position.

If you have a pattern that's not facing the direction you want it to, your only option is to either remake it if it's yours, or delete it if it is not, as you can't redesign patterns made by other people. Alternatively, you could also make a new, rotated pattern based off of the design of the existing piece by looking at it which, while somewhat tedious, would work for your purposes.