Puzzle of the Week #49 - Pair-of-Dice Regained

I have a pair of dice, which are numbered in an unconventional way. Unlike normal dice, where you can achieve a maximum total of 12, and often the same total can be achieved in several different ways, (eg 5+2=3+4=1+6=7, etc), these dice can total any number from 2 to 37 in exactly one way each.

All of the numbers on the dice are positive whole numbers, and one of the dice has four square numbers on it.

How are the dice numbered?