If something requires a DC input of 15-36V, can I power it by using -12 to +12V to get 24V or does it need to be 0V to 24V? How do I know when I can use -12 to 12 vs a 0V reference for ground?

 

It depends on what circuit ground is referenced to in the supply and the device you intend to power. If it's earth ground, that limits what you can do. If they're both floating, it doesn't matter.

 

Two systems of measure: Dual Polarity and Single Polarity.
While there are push pull special motors usually the magnets take care of that and single polarity is used.

Dual Polarity
In an absolute value problem the concept of length can be illustrated by drawing a line in one direction and another line drawn in the opposite direction. In dual polarity we specify the direction using signed numbers from a reference point. Which is usually zero.
Dual polarity is described like this Positive (+12 Volts) and Negative (-12 Volts) with reference at zero.

Single Polarity
In single polarity we eliminate the negative polarity ( translating the axis by shifting up 12 Volts). The maximum and minimum are now 0 (zero) and 24 Volts. We can boost convert the 24V to 38 Volts this allows for voltage drop. Adjustable voltage regulation is possible but PWM will improve torque 15 - 36 Volts. By adjusting both current (by PWM) and voltage a motor will have less tendency of slowing under load.

The next logical question is: What is the maximum current needed ?

 

Either way. 24V 12V 0V or 12V 0V -12V. You can even do 0V -12V -24V.