In which scene do you think Odysseus best lives up to his epithet as the "man of twists and turns" and why?

Because the phrase pertains to his personality and overall reputation, no single event neatly encapsulates this description. Cumulatively, this reputation is built up through the many instances—each an individual twist or turn—in which he thinks quickly on his feet, makes up elaborate lies, disguises himself, and fools others so that he either escapes or gains some advantage.
One instance that shows his ingenuity is his manipulation and blinding of Polyphemus. Obviously at a disadvantage against the giant Cyclops, Odysseus quickly lies to him, saying only a handful of the men are shipwrecked there. Shut inside the cave, he knows that the men lack the strength to move the boulder away from the entrance, so he must use the giant’s strength. This means he cannot kill him. He is able to blind the drunk, passed-out Cyclops, who then cannot see them sneak out when he moves the boulder to leave the cave with his sheep. The wily Greek almost slips up, however; after telling Polyphemus that “Nobody” maimed him, he later admits that he is Odysseus, which later gets him into further trouble.