OK, folks:
Thanks so much for the help! I have started this thread with the idea of providing the basics of UPS transformer identification in such a way that anybody can use this information to identify the particular unit they have. Since UPS XFMRs come in different "flavors", that may be unrealistic, but so be it. I have attached some photos and my crude drawing of just one of the XFMRs to help the visual learners out there. Note that in my crappy drawing, I have attempted to reproduce both the sizes of the various wires and the colors of their insulation.
Here are the results of the resistance and continuity measurements I made. Please refer to the drawing to match the letters I've chosen to the corresponding wires going into the XFMR. All readings are in Ohms: A-B: 3.7; A-C 3.7; B-C 1.5; D-E 1.8; D-F n.c.; F-G 1.5; F-H 1.5; F-I 1.5;G-H 1.5; G-I 1.5. So far, this is what I have surmised: A, B, and C are the primaries; D, E, and F are one set of secondaries, while G, H and I are another set of secondaries. What puzzles me is that F and G should be electrically connected. If they are two different secondary windings, then they shouldn't connect at all, right? Since I don't know if this unit has been damaged (i.e. shorted out), I don't wish to go on to the next step of applying voltage to the presumed primaries ABC.
A word about my test jig: Elsewhere online I have heard people recommend that a current limiter and/or a fuse be used. If I understand correctly, the current limiter could be a light bulb connected in series with the "live" or "hot" wire that goes to one side of the primary, so that even if a heavy load is connected to one of the secondaries, very little current will flow through the XFMR. Sound right? Some people have suggested that I use a small step-down XFMR (output of 10-12V) to apply a safe voltage to the primary of the XFMR under test, while others have suggested using a variac to gradually step up this input voltage. Since I have a variac, I will use the latter option. The remainder of my test jig is basically a long power cord going into a fused and switched power strip, plus a clear plastic "blast box" which sits atop the XFMR under test. With this setup, I can hook up my test leads to the XFMR with no power applied, then turn the thing on from a safe distance. I haven't posted pics, but this should be pretty easy for anyone to understand and implement.
Update: First, David Hess, SeanB and rstofer, thank you all for your feedback. Somehow the photos failed to upload, but no matter. I went ahead and decided that A and B were the 110V primaries and applied mains power. Then I tested for voltages on the supposed secondaries, and here is what I got: D-E 15.7V; D-F 3.5V; E-F 2.8V; G-H 9.4V; G-I 9.4V; H-I 18.8V. Happy to report no sparks, flames or smoke! Since G, H, and I are the heaviest wires, and since they appear to make up a center-tapped 18.8V output, I think that I will build myself a +/- 12VDC power supply and have done with it. In other blogs I have read about tests that help one to determine the power ratings of unknown XFMRs, but since this one feels pretty beefy and is about twice the size and weight of a 24V@4A power XFMR I purchased for a power supply kit rated at 0-30VDC@3A max, I'm not going to concern myself with the mystery XFMR's power factor. A couple of big capacitors, four big diodes, an LM7812 and an LM7912 plus some other sundry parts, and I'll end up with a nice bipolar supply capable of powering any small audio amplifier I need. Thanks again to everybody.