Normally, I would just try stuff to see what worked, but in this case there's real data that I don't want to lose, so thought I'd ask for advice here first.

I have three partitions on this disk, laid out thusly:

|--root--|--swap--|-------------- hda3 (lvm) --------------|

hda3 is the only one using lvm. What I want to do is shrink hda3 from the front, to give a couple of more (contiguous) gigabytes to swap. There's plenty of room. Here's the output of lvdisplay:
and here's the output of pvdisplay:
So, there's plenty of room; but what I don't understand is there doesn't seem to be any way to tell lvresize or pvresize how to move the lv's or pv's around. In other words, I want to tell pvresize (and probably fdisk, too, right?) to scoot everything over to the right by a couple of gig, then tell fdisk to increase the swap partition by said amount.

I've found several examples on the net, but none demonstrate how to exercise this fine a control over the starting and ending locations, and, as I say, I'd be much more open to just experimenting if these weren't currently in use with (a lot) of real, live data.

TIA!

I don't screw around around with LVM, but I'd (also) be loathe to try my luck with what you're proposing. Better to use the LVM tools to free up some space and just add another swap partition. See the howto on TLDP.ORG for LVM - once the space is freed up, just add another swap. Linux will use more than one quite happily.