Challenge 1 - How to remove the anomalous twist from a telephone cord without working it to the end of the wire.
Mystery 1 - How the bloody hell does it get there in the first place!?
Are you talking about the sort of twist where you can just let the handset hand and unwind, or are you talking about when an individual curl sort-of inverts and switches directions leaving the rest of the cord otherwise unchanged?
If it is the second one, you have to grab it on either side of the inversion and twist it out that way. IIRC, it creates a loop in the rest of the cord, but not a cord inversion, and from there you just rotate the handset once and you should be all done.
The coil of the phone cord creates a twist, you can see this if you grab a section and stretch it out. While it is stretched out, grab the cord with thumb and forefinger of each hand so that your fingers are one to one-and-one-half stretched-out-coil’s length apart. Now, without letting the cord slip in your hand, twist in the opposite direction of the rest of the cord and you will put one of those twist inversions into the cord. To remove, locate and isolate the inversion and give it the correct twist. It’s as easy as that.
deja vu.
To answer this strangely familiar OP - I did it the other day at work. I just unhooked the handset from the phone (it just unplugs) held it up so it could uncoil and then reconnected it.
Sorry, I meant to add that it gets ther by changing ears while talking. When you transfer hands you put a half twist in it. If your job is as slack as mine you can check it out.
I meant the second one js_africanus said (“an individual curl sort-of inverts and switches directions leaving the rest of the cord otherwise unchanged”) Not the kind where the whole cord just curls up. That’s easy to remove.
I read your instructions js_africanus but it seems easier said than done, and I have to get off the comp. When I get to work I will read again and concentrate. If you cause me to put a kink in the line, and then fail to remove it, I will hunt you down and cause you immense pain.
P.S. this thread is not just about phone cords. What other great challenges do we face in these difficult times dopers?
A little pain is a small price to pay for sharing knowle…oh, wait…immense pain…uh-oh…
I can’t think of any good mysteries/challenges. Only joke ones like: where does the white go when the snow melts?
Into the clouds? Actually that’s sort of true as well. Not sure where the grey in grey clouds comes from though…
How do you get chewing gum out of your hair without cutting it?
And how the hell did you get chewing gum in your hair, anyway?
From blowing a big blimp of a bubble, silly! 
To get chewing gum out of your hair, my “household hints” book suggests either SprayN’Wash or peanut butter. Essentially, you need something oily.
And then a shower and some really strong shampoo.