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I have a double gang box that has two single pole switches- each
switch controls one overhead light fixture. The switches were nasty
looking so I decided to swap them out. Unfortunately when I removed
them they just disintegrated before I could look at all the wires.
Now I am stuck trying to figure this circuit out.

Two 14/2 romex cables come into the box. Grounds are tied and
pigtailed- but not any of the other leads. However, there is a two
inch long white wire that was used as some sort of "jumper" between
the two switches. It's looped at both ends, so I know it was attached
to the switch's side terminals, as opposed to the rest of the wires
which used the switch's "push-in" terminals.

Fixture #1
One romex cable runs straight up to this light There is continuity
across the white and black as long as there is a bulb in the fixture.
I removed the fixture and verified there are no other wires up there.

Fixture #2
The other romex has a hot white and a non-hot black. The white lead
is not marked. If I connect a switch across these two wires, the
fixture that was controlled by it (before I messed everything up) will
work. I can't (easily) get at this fixture to see what other wires
may or may not be up there.

I can't understand how this was all setup to work properly & within
code. Assuming Fixture #2 has a neutral somewhere else besides that
box I am working in, where is the neutral for the other fixture?

I think you've got it figured out. It's ass backwards and wrong!!!
You're hot white and non hot black are fine. White brings hot to the switch
and black returns it to the fixture. Then they jumpered the hot to the
second switch and returned it on the black of the second cable. The mystery,
as you described, is where's the neutral coming from? It probably isn't.
They probably attached that white to the box or the ground wires to make it
work