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You'll need to tinker with this.
Looks like 10.6 or 10.7

On those, without proper EFI update you won't be able to use Recovery (remind me what Major Payne said about legs)
But with external bootable DVD (yes) you would.
 
My boss acquired a 2011 iMac, A1312, and I can’t get it to boot into recovery. The previous owner has lost their login password. Every key combo I try results in the iMac simply booting to the login screen (R, Command+R, etc.). The keyboard is recognized and works (using a Logitech with Unify receiver, so not a Bluetooth issue).

Any ideas? Also, does anyone have a clue what version of OS X this is (blurred to obscure the username). The Apple logo has the skeuomorphic reflection. It’s obviously pre-Sierra.

Easy peasy. Create a USB (DVD) installer of High Sierra (on another machine) and boot with it.
 
"The keyboard is recognized and works (using a Logitech with Unify receiver, so not a Bluetooth issue)."

Does the Logitech keyboard have the capability to connect via USB?
If so, TRY THAT (instead of the unifying receiver).

If that doesn't change anything, try a different WIRED USB keyboard.
Even a PC USB keyboard might work (you may have to experiment to find which key to the left of the spacebar is the equivalent of "command"...)

If you try the above and it STILL doesn't work, I'm wondering if perhaps that Mac DOESN'T HAVE a recovery partition on it...?

If it doesn't, it might still be able to boot to internet recovery.
Have you tried that yet? (not the same as "the recovery partition"):
Command-OPTION-R
at boot.
 
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