The Twisted Picot, can be used has a decorative element as well as for joining or integrating your next round.

Let me start by saying if you have been having your picots twist on you and you really didn't want them to. It is mainly due to your thread being twisted as you Tat. Always make sure if your ball thread is twisted to un-twist it. Put a band around the ball and stand up and let it drop and un-wind, or if it's just a little then turn the ball to fix it as you Tat, and the same goes for your needle/core thread.

Now to get your picots to twist in most cases you would use a crochet hook, or the end of your needle to twist the picot as you join onto the previous round. Always making sure you twist in the same direction as you work.

Here is an example of some twisted picots that I made purposely to show you the effect of twisted picots by twisting my thread just before making a picot and this way I can eliminate using the crochet hook each time to make them twist.
One is to show how decorative twisted picots look, especially if used on an outer round of a piece.
Two, this is a easy way to automatically get your twisted picots for decoration and for joining on rounds.

Here you see a line of Twisted Picots made by twisting my thread toward me a couple times, just before making the Picot. Make sure you twist it so the thread does not un-twist itself, if the thread starts to open and split, turn it the other direction. Always making sure to turn the thread in the same direction as you work. As you slide your picot up the twist will automatically happen. See the first three the thread was twisted three times, the next three were twisted twice.

Idea's on how to use a twisted picot:
Let's say you are making a ring of 3ds - 3ds - 3ds - 3ds. Then you could make your center picot a bit longer for a decorative effect. But this does depend on if there will be a join or not.
If you are joining on that center picot then you can still make it a bit larger and then twist it one or two times for the twisted effect and this will take it down to the normal size for joining.
And of course on the last round you can use the twisted picot for a more decorative effect.

I haven't seen much around about the twisted picot on the web, but I thought this was interesting since we are talking about decorative picots. This is from the On Line Class by Georgia Seitz, on A Study of the Picot in Tatting.
Here you'll see how using long picots can make a very nice effect by using long picots to make a woven pattern. And of course these very long picots could also be twisted.

Well I just found an article about twisted picots. This is a good article from Gillian Buchanan, Tatting Editor on Bella On Line.