I have three questions stemming from this. First, is it supposed to be this way? I have a hard time believing it is, since in my research over the past two days it seems that (apart from phone systems) positive grounding hasn't really been used since the 1960's. Second, if it's not, could it be damaging the electronic components of the trailer? My guess is that at some point the previous owners changed the battery and messed up when reinstalling the battery connections. We're probably going to be replacing the battery and some of the electronics in the near future, should we go put the ground wire on the negative post?

Before you do anything, check the power panel to see how that is wired. If you reverse the connections on the battery, it will blow the two 40A fuses to protect the converter from a reverse polarity situation which it looks like you have. You could try disconnecting both wires and connecting your solar charger to the battery the proper way.

Mine is white=negative, red=positive. I have a disconnect switch wired in so I don't have to disconnect the batteries (dual 6V) if I will be without shore power for storage or service. It is on the negative between the negative terminal  on the second battery and the white wire. The red is on the positive terminal of the other battery and there is a jumper between the negative of the first battery (red wire) and the positive of the second battery (white wire) so my dual 6V batteries give me 12V.

I would disconnect from shore power and verify that all the 12V systems work properly and that it all is working with the + terminal grounded. If that's the case, then your system was reversed somewhere alone the way.

Not only would all of the interior polarity-sensitive components need to be reversed (converter, furnace, fridge electronics etc) but it makes me wonder how in the heck they ever got the trailer auxiliary charge line to work, or if they disconnected it, or maybe, the trailer was hooked up to an old positive-ground classic pickup? 

Or maybe they somehow electrically insulated the hitch ball from the frame of the tow vehicle?

Make sure you use a DVM to check things before you make any changes.

I'm disappointed that the OP hasn't followed up.  If he found out what was going on, I think we'd all like to know.  If he didn't, we'd be interested in continuing to troubleshoot.

TT

If they did that when the battery was healthy and fully charged, it probably damaged the converter or at least blew some fuses. The electronic control boards in things like the fridge and the furnace and water heater might be fried as well.

If the battery was completely dead, and if the cells are dry, then the battery is 'open', so when it was hooked up just for a quick driveway sale, probably nothing was harmed, but nothing would work.

Yes, I agree.  You can see from the pics that indeed the + battery terminal is connected to the white conductor, which going to the frame. So yep, I would expect that at the very least there would be open fuses in the charger circuit as those are there for reverse polarity protection. I've seen some comments on other sites that when folks have accidentally reversed the battery connections those have blown and protected everything else so maybe that's all that would happen,  but that is something I don't personally ever want to find out .