a Turing reduction
/ TYOOR-ing /
A mapping reduction is a strict, one-shot translator: translate the question once, ask once, report the answer unchanged. A Turing reduction loosens every one of those restrictions. Picture being allowed a magic hotline to an expert on problem B. You may call as many times as you like, with any questions you cook up, and combine the answers however you wish, including flipping a yes into a no. A Turing reduction says: if I had that hotline for B, I could solve A.
Formally, A is Turing-reducible to B, written A <=T B, if there is an oracle Turing machine that decides A while making queries to an oracle for B, a black box that instantly and correctly answers 'is x in B?'. Unlike a mapping reduction, the machine may ask many adaptive questions (later queries can depend on earlier answers), use the results in any logical combination, and negate them. The transfer rule is the natural one for decidability: if B is decidable and A <=T B, then A is decidable, because you can replace the oracle with B's actual decider. Read in reverse: if A is undecidable and A <=T B, then B is undecidable. So Turing reductions still spread undecidability.
Turing reduction is strictly more powerful than mapping reduction, and that extra power has a cost. Because a Turing reduction can negate the oracle's answers, it does NOT respect the difference between recognizable and unrecognizable: A_TM and its complement are Turing-equivalent (each easily solves the other given an oracle), even though one is recognizable and the other is not. That is precisely why, when you want to separate recognizability classes, you must use the weaker mapping reduction, whose yes-to-yes faithfulness preserves the distinction. Turing reductions give the most general meaning of 'solvable relative to' and underpin the structure of the uncomputable, including the arithmetical hierarchy and the notion of relative computability.
The complement of A_TM Turing-reduces to A_TM: to decide 'does M NOT accept w', ask the A_TM-oracle 'does M accept w' and return the opposite answer. This single negating query is illegal for a mapping reduction but perfectly fine for a Turing reduction, showing the two reductions differ.
An oracle hotline for B, queried adaptively and possibly negated, decides A: that is A <=T B.
Turing reductions transfer (un)decidability but NOT (un)recognizability, because they may negate the oracle's answer. To separate recognizable from unrecognizable languages you need a mapping reduction, not a Turing reduction.