I have Acrobat Pro and find that sometimes if I add a page from one file into another, any comments that are copied to the new page are automatically rotated if the added page was not the same orientation as the existing document, even if I previously rotated it and saved it to match.
How do I either stop this happening, or how do I rotate the comments to their original orientation?
You cannot stop this from happening. Here is why you see this behavior: Just because you see a page being oriented "correctly", does not mean that the page was actually created with that rotation. Depending on the application (and the PDF converter used), a page may be created with it's content actually rotated by 90 degrees, but when displayed, you would have to turn your head to read the content, so the application/PDF generator also adds a rotation flag to the page that says "rotate by -90deg" and this way, the PDF viewer will make sure that you can see the page in it's natural orientation.
Whenever you place an annotation (that is placing it for the first time, not coping an existing one), Acrobat will look at the page rotation, and then place the annotation with a rotation setting applied so that it looks correct on the page. When you copy that annotation, you are also copying that rotation setting so that when you place the annotation on the same page (or a different page in the same document that was created the same way), it looks correct again. However, if you paste that rotated annotation on e.g. an unrotated page (or a page with a different rotation setting), you will end up with something that does not look correct.
What can you do about it? Unfortunately, with the tools you have available in Acrobat, there is nothing you can do. You can write a script that rotates a selected annotation, but this would have to be done after you place the annotation.