Running LEDs from an AC supply
There are several options for operating LEDs from an AC power supply. Many stand-alone LED fixtures simply have a transformer between the wall socket and the fixture to provide the required DC voltage. A number of companies have developed LED light bulbs that screw directly into standard sockets, but these invariably also contain miniaturized circuitry that converts AC to DC before delivering it to the LEDs.
Another approach is to configure the LEDs or die themselves into a DC bridge circuit . Although AC is input to this LED bridge circuit configuration, the LEDs are still driven with DC and this approach requires more drive power than a "true" AC-LED design.
Board-level C3 LED device
One early form of a "true" AC-LED system, in which the devices operate when connected directly to an AC supply, is the "Christmas tree light" approach. Here, multiple LEDs are connected in series so that the voltage drop across the whole string equals the supply voltage.
However, attempts have been made to develop "true" AC-LEDs at the assembly or packaged device level.. At the forefront of these developments are Lynk Labs, Seoul Semiconductor and III-N Technology.
The technology developed by Seoul Semiconductor and separately by III-N Technology uses the Christmas tree approach at the die level. The AC LED device is actually made up of two strings of series-connected die, connected in different directions; one string is illuminated during the positive half of the AC cycle, the other during the negative half. The strings are alternately energized and de-energized at the 50/60Hz frequency of the AC mains power source, and thus the LED always appears to be energized. The technology developed by Seoul and III-N specifically relates to LED devices designed for high-voltage 50/60 Hz mains AC power.
If it doesn't use a transformer for lowering the supply voltage there's a very good chance is does work from DC. Most wall-warts will run from a DC input because (like your LED panel) they have a bridge rectifier that converts ac into dc. Bridge rectifiers pass ac or DC exactly the same - they are just steering diodes that always push the most positive input voltage to the +Vout and the most negative input voltage to -Vout
It did not blow up (or not-work or get too warm or short the supply out) because it doesn't use an input voltage transformer (as per the answer in the paragraph above).
When you powered it from 12V dc I suspect that there isn't enough voltage to allow all the circuits to work correctly. AC 24V will produce a peak voltage of about 34V and this will likely feed through a bridge rectifier to give about 32Vdc - a lot higher than 12V dc you fed in and this I suspect is the main reason.
Ditto for AC 12V - it has a peak voltage of 17V and after a couple of diode drops in the bridge will settle at 15.6VDC - if you had a bench supply and fed it with 17Vdc I'd bet it would work.