Thanks to everyone for the most excellent advice.
I worked on it yesterday and while it's not quite indistinguishable from new, it's definitely getting there. It's at least close enough to imagine new. I still need to locate some of the finer grades of abrasives mentioned by Bruce (I'm mising the 1500-3000 grades of abrasives). The finish is currently a sort of polished satin - you can see your reflection, but if you look carefully, you can still see some fine texture and the reflection is a little milky rather sharp and clear like the reflection from an actual mirror.
Here are the results so far along with a few lessons learned along the way.
Lessons:
1) Use the right abrasives
I started with jeweler's rouge and found that after an hour of polishing, all I had succcedded in doing is to round the edges of the scratches. Bruce was absolutely right (no surprise there) - the simple polish was useless on scratches . There's no substitute for a complete set of graded abrasives. Having the right abrasives also saves a lot of time.
2) Work slowly and carefully
Since I was lacking a good intermediate grade of abrasive, I thought that I could make up for it with a finer abrasive and some "elbow grease". The result: during the frenetic rubbing, a particle of grit found its way between the cloth and watch and I ended up with a big ol gouge mark. Back to square 0 (or -1).
3) Have a clean, organized workspace
A tidier workspace could have possibly prevented the grit particle problem cited above.
I struck out yesterday in locating fine enough sandpaper, but I'm hoping that if I can find some 2000-3000 grade, then I can eventually get to that mirror finish. I still have a hard time imagining this taking less than a few hours, but I'm guessing that with the right tools and experience, it will get a lot quicker.
Again, thanks for the help.
-abe.