I've looked a few times in different sections of the tiny84 and mega328P datasheets, but I can't find an explicit answer to this.
Are there voltage/frequency limits for the Low Power Crystal Oscillator, or is it capable of running at any frequency and voltage within the recommended operating conditions? The Full Swing Oscillator has a note in its section (helpfully buried at the end of a paragraph instead of in a proper table) that says it only works above 2.7V, which the Low Power one doesn't have. The assumption would be that the Low Power one doesn't have that limitation, but I don't like making that assumption. That's the kind of thing that leads to unpredictable and inconsistent behavior, the worst to debug.
The scenario I'm thinking of would be if I wanted better accuracy than the internal RC oscillator, I've got a lot of 16 MHz crystals. Instead of buying a lower frequency one, hook that to the micro and program the CKDIV8 fuse so it starts up at 2 MHz by default. That way it can run all the way down to 1.8V.
Does that work?