So the main power comes through a relay, controlled by the ignition switch?
You could try by-passing the relay with a fused live supply direct from an un-fused power source. Yes, when connected, the radio would be continuously on, but if it works as it should this way, I'd suspect the relay. Or just go ahead and substitute a known good relay.

Yeah, it looked like the 12v power comes from the aux relay, and the power for the radio memory is the same as the interior lamp. I just didn't think that would've been enough power to actually run the radio. And if that were the case, wouldn't it just not turn off, instead of having the switch backwards? 

Not sure about your unit, but modern radios typically have a main power feed controlled by the ignition switch and another smaller feed that is always on to provide enough energy to keep the memory of which stations you have programmed in. The unit is also always grounded. Maybe if the ground is not good, the power is back-feeding somehow. Just a guess.

Seemingly at random, my Mini decided to run electrical power to my radio in reverse. What I mean by that, when the ignition switch is off, the radio is on. When the ignition is set to accessory, on, or start, the radio is off. I'm trying to troubleshoot what the issue could be, and was hoping for some pointers. The radio is an aftermarket unit, and unplugging it/ plugging it back in didn't help, nor did disconnecting the battery. Nothing behind the radio appears to be frayed, and from what I understood of the wiring diagram, the other electrical loads off the same place (heater, windshield wipers) all work. I've considered it might be the ignition switch, or something internal to the head unit, but any other opinions would be greatly appreciated!