DC Motor used in cordless drill - is it polarity-sensitive?
All permanent magnet motors are polarity sensitive. Some DC (also called universal) motors have a stator winding rather than a PM .. they run in the same direction regardless of input voltage polarity. To reverse that kind of motor you have to reverse one of the windings. The latter kind of motor is not very common in cordless power tools.

Edit: Given the additional information that this is a Dewalt drill, here is a typical exploded diagram. 



We're getting off-topic here, and well into repair territory, but looking at the diagram you can see the gearbox/clutch (14), the motor (13), which appears to be a brushed permanent-magnet motor, and the switch-speed controller assembly (9). Applying DC voltage to the motor (13) leads (disconnected from (9)!!- "stripped out" as you said originally) and it should run forward or backward, unless the brushes are worn out or it's otherwise damaged. 

Applying reverse voltage to the input of what DeWalt sometimes calls a "VSR Switch" (9) will likely damage that module- there's a lot more than a "switch" in there. Applying voltage of either polarity to the motor with it still connected to the output of (9) might well damage the module. As Russell says, the default in design would not be to add cost to every unit in order to protect against this sort of thing. 

Brushes are cheap (maybe $5), and they do wear out (with similar symptoms to what you describe), so maybe that was all that was wrong (is wrong) with the motor, so perhaps you need new brushes (and now the VSR Switch (maybe $50)). 

Video on replacing DeWalt motor brushes: http://www.youtube.com/watch?v=tECeZi6IfvU

The VSR switch (9) is a non-repairable module integrated with the trigger switch.