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Is there any way (plugin/script/compound/voodoo) to grow (and keep) hair only within the confines of other geometry,
i.e. to fill a (hidden) geometry object (and it only) with hair?

It depends on the effect you want, if the hairs have to move perfectly with the volume during deformation it may be complicated.
Check out kristinka cut by volume, I don't know if it has the option to emit strands in a volume but it shouldn't be to difficult to setup, just delete the particles that are outside the volume.

Thanks. I take it this won't be possible with native XSI "Shave" hair, then? ;)

There's 'cut map' in Shave, which re-samples hairs along hairs, to some smaller length. Effect is visible only on final, hair filler, not on guides. But, as name says, it's relative to actual length, controlled by weight map, not by external geometry.
Could be an exercise, let's say to get length of Shave guides, by Melena or something, then to compute relative length of hair part, occluded by external geometry, then, to apply result back to weight map (I think process should be frozen at some point, to avoid dependency loop). Kristinka Hair 'Cut by Volume' uses ICE 'Point in Volume' node to find how many members of strand array are inside volume, so that's base for new relative length.
All that, only if you want a plain cut - like Star Trek Vulcan's hairdo, navy cut, medieval hairdos from Prince Valiant comics or such.