I'd start by looking for a datasheet for it.
After that I'd probably hook it up on the bench, an ohm test well tell me the current draw. Then pick a resistor (to limit current) and apply 12V in one direction, then in the other. In one direction it should get shunted down to 0.7V. The other direction should look like a resistor divider.
What'd be ideal is a current source and testing in both directions, but I don't have one so I wouldn't use one as I wouldn't want to waste time making one...
A DMM might not work well with a shunting coil. Let's see:
My Fluke 179 measures 84Ω on a typical "big" auto relay, unaffected by a 1N4004 shunting it in either direction. Diode check fails, due to the "short" of the relay. A DS2E measured 350Ω and when I put the same diode across it, in the diode mode it drops a mV when I change directions. A G6K measures 1.29k and here the diode checker works, showing 1.29V in one direction, 570-something mV in the other.
A cheapo HF DMM (their cheapest red one) operates identically. If you pull out some ancient meter, all bets off, if it applies more voltage/current to test with. But otherwise, if the coil resistance is under 500Ω or so, diode checker likely won't work, and it'll be hard to tell.
Anyhow. Those diodes are apt to be stupid robust, and if they die, it's not the rest of the relay that is apt to die, it's whatever is driving it. If the coil opens up, a continuity test should find. But I'd think it was the contacts that would give up first.