I can confirm what PhillipZ said, and give abit more information.
Firstly, the behaviour we see was intended to be a feature (although I'm not convinced of its usefulness): whenever you insert a USB connector into the TyTN's plug, it will sense the +5V. It then "looks" at the data lines to check if there's a real USB device (like a PC or a USB hub) at the other end of the cable. USB devices must first negotiate with the device at the other end of the cable about how much poer they're allowed to drain. If the TyTN senses that the data lines are dead, it tries to be a good citizen and does not draw any power from it.
Unfortunately, none of the chargers are USB devices, so the data lines will always be dead. To allow the TyTN to still work with such a charger, it was decided that the TyTN will also look at pin 4 (the one PhillipZ referred to as number 2; the official USB standard counting starts from the other end, so it's pin 4 really), and if this is connected to pin 5 (i.e., to ground), then the TyTN knows it's okay to start drawing power from the device.
This scheme is not specific to the TyTN. Most (four out of five) mains chargers with USB connectors I have indeed have a "grounded" pin 4. These come from very different manufacturers, including some Chinese ones. Some of them have a very low power rating (0.5 A), and they still charge the TyTN nicely even when the display is on. So the power isn't really the deciding factor if a charger works or not.
Then there are the chargers (mains or car) which do not have a fixed cable, but just a USB connector, meant to be used with a standard USB cable. These will indeed not work with the TyTN. The reason is that none of the available USB cables do have a grounded pin 4 at the Mini USB end. And they mustn't have that: pin is marked as "reserved" in the USB standard, and nobody really knows what all the USB devices would do if connected to a PC via such a cable. Some of them might do nasty things. So the cable manufacturers play by the rules and leave pin 4 unconnected.
This is of course no concern for chargers with a fixed cable, as the manufacturer can be sure the cable will never be used to connect two real USB devices. That's why such chargers are likely to work - for me, four out of five.
The HTC Magician (MDA compact, XDA mini) originally showed the exact same behavior. It was later modified with a ROM upgrade, so that the Magician could be charged even on charger with unconnected pin 4. I find it a bit annoying that HTC reverted back to the old behavior - it's really a feature that no-one likes, and I never heard of problems with the Magician, although technically it may be violating the USB standard.
I verified the pin 4 issue with a special cable I made (back when I bought my Magician; sacrificing the Magician charger). This cable has now a normal USB plug on one end, and a Mini USB plug with pin 4 grounded on the other. Using this cable, all my car and mains charger without a fixed cable could charge my TyTN with no problem. I have never dared to try and use that cable to connect my PC and the TyTN...
The fact that the TyTN can be charged with an unconnected pin 5 charger immediately after a soft reset seems to be a bug. Interestingly, software seems to play a role here: with an O2 ROM, it was possible to charge the TyTN exactly once. If you disconnected and reconnected, i wouldn't work until the next soft reset. With the current South African ROM and no programs installed, charging worked as many times as I wanted, but only until I connected the TyTN to the PC. Once it sensed a real USB device at the other end, it refused to charge from an unconnected pin 4 charger.
One last thing: on the three Mini USB connectors I cut open, the pin 4 metal clip was considerably shorter than the clips of all the other pins. It was not possible to solder a wire to it without cutting very deeply into the connector. So, unfortunately, it's very hard to modify a standard cable so that pin 4 is grounded. It seems that until (if ever) a ROM update changes the behavior again, the best thing to do is get a charger with a fixed cable which specifically supports the TyTN (or Magician).