I have a DPDT switch that returns to the neutral position. What I want is to be able to wire up a light that turns on and stays on when I push the switch to the "forward" position, then stays on in the neutral position, then the light will only turn off when I push the switch to the "reverse position", then the light again stays off in the neutral position. Does anyone know how to do this?

You can do it with electronics.
A simple way would be a latching relay.
May be a problem or not, but you could move the switch OR reverse(up or down) to turn to the opposite condition. You could actually have just a spst momentary pushbutton for this. With a dpdt, maybe even a triple pole double throw switch, then momentary down would be off, and momentary up would be on.
Some remote controls have a power button, no on and off, it just toggle between the 2 conditions, with 1 momentary switch.
Can you tell more?

I'm using the DPDT to control a motor where I need to reverse polarity to change direction of the motor. However the motor only stays on for about 2 seconds in either case, this is why I'm using a momentary DPDT that returns to a neutral position. A single coil latching relay sounds like a good solution, I think I'll experiment with that. As for your other suggestions I'm not sure I fully understand how to implement them in conjunction with the momentary DPDT, however thanks for the latching relay suggestion.

If you are needing only 2 seconds, you may not want a latching relay.
What I might do and since you mention only 2 wires for reversing, I take it you mean a DC Motor.
On your dpdt switch connect the 4 outer terminals making an "X" . The 2 center terminals can be hooked up to your motor. Then apply your dc voltage to either end of the "X".
Center will be off and will go forward or reverse for as long as you hold the switch.

Yes it's a DC motor and its currently already wired to the DPDT switch as you described. Now I just want to add the light as simply as possible so that "forward" turns the light on (which then stays on in neutral), while "reverse" turns the light off (which then stays off in neutral). I thought the single coil latching relay sounded good (for controlling the light only), but why then do you say I might not want one?

You could use a diode off switch so it will only light during forward, and will be off in reverse and off condition. Or an Led and a resistor. They only conduct(light)one way.
I was saying a latching relay may not be desired if only for a couple of seconds. You would have more precise control with releasing the switch?
What type of light? you are not limited to DC Lighting, a diode and a relay/contactor could control Line voltage or any amount of lighting.

Edit, I see you want it to stay on in neutral, an SCR could work.

The Silicon-Controlled Rectifier (SCR) : THYRISTORS

Scroll down to the SCR Testing and the circuit below showing motor or lamp control with a NO switch to start and a NC switch to stop.