My Woofer Sounds Like a Tweeter

I just pulled out a pair of mid-80's vintage Polk Audio bookshelf speakers from the spare bedroom and I wired them into a second system.

One of the speakers only produces the high frequencies from it. I opened the inside of the speaker enclosure and checked if any wires were loose or broken, but everything looked OK.

The tweeter seems to be operating normally, but the woofer is only radiating high frequencies.

These are not very costly speakers.
What is the problem and is it a worthwhile fix?

Could be your crossover is not delivering the goods to the woofer and only delivering high-frequency signals (some of which are getting to the woofer). Just a guess, but that's where I'd look first. I'm assuming the obvious, that the driver itself is in tact and does not have any obvious damage to it.
See if the woofer cone can be moved easily when you push on it. It may be possible that the voice coil has frozen.
I would assume the woofer is toast before the xover is fried. Theo's advise is good, but not sufficient. If you can, remove (or just disconnect the woofer from the xover and connect it up directly to the amp (at low volume). If if sounds like a woofer, it's the xover. If it sounds like a tweeter, or nothing at all, it's the woofer.

Looking forward to continuing the diagnosis...
Actually Dan is right but you can actually bypass the xover for a quick test. Just clip some jumpers on the input of the xover and jump directly to the woofer connections. As Dan states though, low volume.
I will give these tests a try and report back with the results.
Theo is right - disconnect the speaker terminals from the speaker wire. Then just touch the speaker cable directly to the woofer (at low volume and don't let the speaker wire tips touch each other). Can't get any simpler than that: no jumpers required.

Theo - your turn :)
I am all in, my brain hurts...
I am all in, my brain hurts...

Uh oh; D.P. Gumby's stereo advice hotline...smoke and explosions are sure to follow.
Nah, just:

1) put on some music at a low volume & turn off the amp
2) disconnect the main speaker wire from the main speaker terminals
3) make sure the speaker wire tips don't touch!
4) turn on the amp (repeat step 3)
5) touch the speaker wire ends directly to woofer terminals inside the speaker (repeat step 3)
6) listen: bass or no bass (repeat step 3)
7) turn off the amp (repeat step 3)

See mom - no smoke!
3) make sure the speaker wire tips don't touch!
4) turn on the amp (repeat step 3)
5) touch the speaker wire ends directly to woofer terminals inside the speaker (repeat step 3)

Just an addendum to Dan's advice...

If you have a tube amp you may want to touch the speakers to the woofer terminals before you turn the amp on. It's not the best thing for a tube amp to be receiving a signal from your source with no load on the speaker wires.
Just use a couple wires and a battery(ie: AA, C, D, or 9V), briefly touching the woofer terminals. You can do the same test with a multimeter, set to "ohms". The speaker should either push out, or pull in, depending on the polarity of the hook up. If it speaker has ever been stored in an area with high humidity: there could be rust in the voicecoil gap. If it's ever been dropped: the polepiece or magnet could have shifted. Either would hold the voicecoil captive, and eliminate it's excursion. If the voicecoil is burned: you'll get limited excursion, and a lot of rattle(with music).