If the LEDs are desoldered, then the legs are cut... so there is no leg that is longer than the other.
"Reversing the polarity" may sound like something out of Star Trek, but it is as simple as to swap + and - .
If the LEDs are somewhat transparent then you will be able to see the innards if you have good eye sight. Those innards are often a bit asymmetric and once you have figured out which legs are + and - on one LED in the pile then you should be able to find the polarity on the others as well.
Or you could build a LED tester. Connect a 3V battery pack, a resistor and two empty terminals in series. Then test the LED against those terminals to figure out which direction works. The resistor is necessary to not burn the LED.
I dunno about the value of the resistor though ... You could check a web site that sells LEDs what the LEDs are rated at, then google for a "LED Resistor calculator".