12 V DC is a given as you say, since these were often intended to be used within cars. And since this is a rather recent device, one could almost safely assume the polarity of the barrel jack would be center positive - as mostly older devices dating back up into the very early 90s would sometimes use center negative.
Just out of curiosity, is there any kind of indication on the label that says anything about the current that the device demands? I wouldn't be surprised if it ranges somewhere between 1.5 and 2 amps at 12 V.
If you have calipers at your disposal, you could also measure the diameter of the socket and the inner tip.