iPhone Mail app + Gmail passwords = No longer works

Thanks to (fill in your favorite explative) Heartbleed, I've spent the entire day resetting passwords on dozens of sites / accounts, between work and personal, including two Gmail IMAP accounts.

Afterward, I went to my phone, Settings > Mail, replaced the passwords, then...got this:

Cannot Get Mail
The user name or password for "imap.gmail.com" is incorrect.

I logged into both accounts on the MBA, via Chrome. Worked fine.

Tried them in the (wildly inconvenient) Gmail app on my phone. Again, they work fine.

Same for Mail on the MBA. The new passwords are definitely correct.

Here's what else I've tried:

1. Re-entered both the user names and new passwords into the phone's Settings > Mail panels...ten times

2. Powered the phone all the way off, then back on

3. Deleted both Gmail accounts from the phone completely, then re-added them

4. Had someone else delete the Gmail accounts, then re-add them

5. Deleted both accounts from the phone (again), then re-entered them using the "Other" button instead of the "Google" button provided during the set up process

6. Deleted both accounts from the phone again, then manually removed the existing Gmail servers from the Servers list. Powered phone off/on, the tried adding accounts back into Settings > Mail.

Nothing works. I'm so far past p****ed, it's not even funny.

Anyone have some insight into this fiasco?