If a 6x6 or 4x4 wants to twist, it will do so and you can do little to stop it. However you can decide which end will twist!
Buried pillars
If you bury the 6x6 into a concrete pillar, the bottom will be fixed and the top will move, perhaps before you get around to locking in the rest of the deck framing. What I do is to dig my post holes below the frost level. Then I pour about 3 inches of concrete into the bottom of the hole, flatten it out and let it set. Then I sit the 6x6 on top of this footing. The rest is filled in with soil, not concrete. Most decks that are not high off the ground or on hillsides simply do not need all that concrete we pour under them. If you build the top framing immediately, even before filling in the hole, then the top of the 6x6 is locked into position and if it wants to twist, it is the bottom that moves, causing no trouble to anyone.
Fixed connection concrete to wood
If you are putting StrongTie hardware into concrete that does not have the capacity to swivel, place the beam first, suspending it in the air if necessary with the 4x4 post hanging from the beam and the lower connector dangling where there will eventually be concrete. Then pour the concrete. That will guarantee that the attachment hardware is perfectly in line with the deck, rather than trying to force the deck structure into poorly lined up fixtures.
Adjustable connection concrete to wood
If your concrete to post hardware is adjustable, then it will simply swivel a bit if it is twisted, or twists in the future. In this case you follow the standard procedure of pushing it into the wet topped off concrete column -- the swivel allows for adjustment after the concrete has cured.
Trimming the twist straight
If you have a 6x6 that has twisted on you and you need to bolt structural beams or joists to the sides of this post, simply cut minimum shoulders on both sides that are parallel to the line of run of the wood to be attached. You won't remove much strength from the 6x6 and everything will be solidly attached and in line. The extra strength you get from a good fit is more than you loose by cutting a bit of wood.
Follow this link for more information on Basic Construction connectors.