How do I rotate the this image by 30 degrees but still have it be tileable?
I was going to say "you can't", but apparently it is possible:
I'd rather not do all of that and come up with something else.
You can use Illustrator Pattern Maker.
Here is the demonstration of what I made quickly, I could have adjusted it but I guess it gets the point across:
You can see the detailed instructions in this Youtube video.
This is why it's important to know what other programs can do. For example, there are things that only Corel draw can do which Photoshop and illustrator can't, even though they look very basic.