Nikki,
I would wash them carefully, trying not to damage the surface. When they are washed trim away any small roots ( which won't help anyway) and at that point you may be able to see a way to untangle them. I would recommend that. Let them dry again, dip them into a fungicide powder, wrap and store. Good luck.

I always wash them before drying and storing, no matter what method of storage or in what form: whole clumps or separated tubers. My clumps are pretty muddy and wormy when they come out of the ground, and I want that stuff off before storing.

Yes, those tightly bunched tubers can be tricky. Like Jroot said, get rid of any extraneous roots and stems to clear your way for dividing, if you're going to. Try to spot good tubers with strong necks and possible eyes. If you see dud (eyeless, broken or weak necked) tubers, cut them out of your way too; that will make it easier to divide and harvest wanted tubers.

I found a little sharp narrow-bladed tool in the pruning shear aisle called a "fruit secateur" or something like that- the packaging is long gone. They are great for fine trimming into stem material and also for making a narrow stab into a big mess or where a more delicate cut is desired for the very situation you describe. I am terrified of a knife, although I've used those too. I only have ten fingers and I'd like to keep it that way! Another option is to cut off most of the big stem, then hammer a chisel right down the middle of it and let the chips fall where they may.
You will lose some tubers no matter what you do, so why not experiment around with what works best for you.

How do you normally store your tubers up there in BC?

Storing way up north here lol. Well, my first year I stored them in paper bags (as I was told to do) and I had great success. Last year I kinda got lazy and tossed them in plastic grocery bags (when I dug them up) and had ALL intentions on putting them in paper (so all the Moms around here told me to do ) but I never made it that far. This year....I'm doing alot more (call it safety) After heaps of reading I had better do it properly!! :-)

I'm going to take my tubers down and wash them up tomorrow and see how things go. I'll see if I can take a pic of my mass of mess. For some reason I think unless I'm cutting right thru (like cutting a potato down the middle) I'm going to have to leave them alone.

I have 1 that has 2 stems that are twisted around each other (seems very delecate) and another that has 2 stems but all joined to the same "ball".

Am I the only one that has really super deformed looking tubers???

They'll get worse if you replant the whole gnarly mess next year. Some clumps are just tightknit like those you describe.
All this talk of clumps has me mentally bracing for the big dig, coming right up in 4-6 weeks I'd guess.