I would like to run some new outlets in my garage. There is
currently a subpanel located there, the run will be fairly short. I
have a section of armored cable which was left over from work at my
office that I'd like to use. I'm thinking of using a setup which I
have seen used in other houses, but want to run past you first.

The armored cable has two 10-gauge (white & gray), 4 12-gauge
"hots" (various colors), and two 12-gauge grounds. I'd like to use
it to run 4 20-amp circuits. Circuits A and B would come off of
opposite phases, and share one of the neutrals and one of the
grounds. C and D would be run similarly. That way, neither neutral
would ever carry more than 20 amps.

I know that I would never exceed rated ampacity for any conductor,
but I do not know the NEC well enough to say whether that will meet
current code or not. Anybody want to chime in?

For code questions, only your local code enforcement office can give you
the last word on this. Often they make changes/additions to the NEC and
that is what you'll have to work to. The NEC is MINIMUMs: Local offices
often add to them.
Give them a call; they're usually more than happy to discuss it with
people. You can also make certain of the permit situation at the same
time; you may need to pull a permit but in your case it'll be a very
simple process as will the inspection.

Twayne