Un-pairing then re-pairing (restoring from backup that had just been made during un-pairing) doesn't help. After the restore I did this:
1) switch watch face to siri
2) open stopwatch, start it
3) return to Siri face, confirm that the stopwatch is running (as it's now the first card on the Siri face)
4) place hand over watch to put the screen to sleep
5) wait a bit, then raise wrist
The Siri face SHOULD show, but instead it goes right back to the stopwatch app.
6) stop the stopwatch
7) place hand over watch to put the screen to sleep
8) wait a bit, then raise wrist
At this point, since the stopwatch is not actually counting up, the watch acts as expected and shows the Siri face (or whatever face I happen to have active as my current watch face).
This must be a bug; the point of the Siri face is to be able to raise wrist and see the various cards, including stopwatch status.
I've moved to another stopwatch app in the meantime.
Perhaps un-pairing then paring as a new watch may fix this, but I want all my existing data and config.