Shame on them for making such a beast! Yes, I'm serious. I consider that to be pretty unfriendly.
If you have access to a current-limited power supply, you can try running the voltage up slowly and notice if the current rises abruptly at too low a voltage. Set the current limit to some reasonably value for the device. I'm assuming/hoping here that it draws relativly little current in normal operation, so you don't have to set the current limit to destructive levels.
Some devices have diodes to protect things: either a shunt diode to draw big current through the diode (and maybe blow a fuse before the diode fails or a circuit board trace burns up) if things are backwards, or a series diode to allow power supply current only in one direction, or a diode bridge to allow the power to be connected either way without mattering.
If the device has terminals (like headphone jacks or line input jacks) that likely have something tied to ground, you could perhaps find if that's connected directly to one of the power supply pins. It's usual, though not guaranteed, that "ground" will be the negative terminal.
If the device has an included battery holder, you may be able to deduce the polarity from finding one end of the battery holder connected directly to one pin of the power connector.
And finally, if you can open the device up and peer into it, you may be able to trace out enough circuit that the polarity becomes obvious. It's common, for example, to have a large polarized capacitor across the power supply, and the polarity markings on it will immediately tell you which way to apply power.
I'm assuming here that you have access to some basic test equipment, like an ohm meter, and basic circuits knowledge--if not, get someone local to you to help.
Cheers, Tom