All my email accounts use IMAP, and it seems the iPhone barely caches
anything. Every time I view a message, it seems to download it again.

Sometimes you get an email that you'd like to save on the iPhone or
iPod touch. For example, it might be an email that you want to see on
a flight, or anywhere you don't have Internet access.

I've found two ways to store the content of an email on the phone:

(a) Reply to the message, then save it as a draft (Cancel -> Save
Draft). Then you access the saved message by going to the Drafts box
in that account.

(b) Copy the content of the email, then paste it into a Notes
document. The limitation here is that Notes can't handle rich-text
documents, so you lose fonts, formatting, tables, graphics, etc.