With a TJ as old as his '97 is, it's likely that if the Tracloc LSD (posi) is present that its clutch pack will be completely worn out by now. If the clutch pack is indeed worn out, spinning one tire will cause the other tire to spin in the opposite direction like an open differential will.
A couple things that might confirm the Tracloc's (the Jeep name for a Posi) presence... a warning sticker inside the glove compartment warning of its presence and/or a small steel plate bolted to the axle's differential cover at the 1 o'clock position indicating that a friction modifier is required.
Otherwise, it will take a look by someone knowledgeable. I have a drawing on my home computer that shows the shape of the two (open vs. Tracloc) cases I can post tonight. Basically, the opening into the case (carrier) is pretty square with an open carrier, with a more rounded opening on one side with the carrier that holds a Tracloc.