On this forum it seems that most people think that the Periapt can turn your opponents dice into your own. However in the group that I play in, I have been told that this is incorrect and that the way it is interpreted is that you can save 1 of your dispel dice and put it into your own power dice in the next magic phase (like a nerfed rod of power). That is always how I interpreted it as well. Am I correct in my interpretation?
No you are not, but you are fully entitled to it. Gav clarified years ago in the White Dwarf magazine that you can save any un-used dice, even your opponent's (as the rules make no exception).
Interesting, what do you do in the case of an item such as the rod of power having a claim on your opponents dice then?
In your given situation, compromise. As each item claims a spare die the pool stays like it is. Don't make it more difficult as it is :-)
In case it is my turn and my opponent with rod has used all his dispel dice and I have one left, I claim it as it is my turn and vice versa.
Hope this helps