[bogiediver]

LEDs can be voltage and polarity specific - or not... depends on how they were made - i.e. what circuitry is included.

For example - there are LED 7" round headlights that are neg ground only and put out full light from 9V to 33V - simply don't work below or above.

A 4" round aux lamp works 9v to 15v - w auto shut off at 18v...

Marker lamps designed for 13.4v (in other words the typical 12v system) only.

All these from the same company...

Some single intensity LEDs can be 'dimmed' with an inline resistor; or with the proper wiring including a resistor in one leg and a diode in the other, can become dual intensity.

I use all the above on my motorcycle (12v neg gnd)...

Most LEDs are polarity specific (and protected), but not all - so you have to be sure to get what you need.

Unfortunately, the correct answer to the true original question (whether the brightness would vary for the same LED on 6v vs 12v) is - it depends on the manufacture of the LED device.

I use a set of 6v pos gnd, tear drop shape, dual intensity, LED clusters mounted to the rear bumpers of my roadster to give me running, turn and brake lights - I feel safer with those in place...

FWIW