A person who twists facts to appear infallible?

Sophism, and by extent, Sophists, are individuals who use logic to twist the meanings of words in order to present themselves in a better light, or otherwise twist logic to suit their needs.

This is the modern day understanding of the word, though it used to carry the meaning of anyone who practiced logical understanding, it has come to mean one who twists logic due to the nature of its abuse.

Today, it holds the meaning of someone who twists logic to their own end.

Consider casuist

one skilled in or given to casuistry

Casuistry is defined as

specious argument : rationalization

You also might consider rationalizer, one who attempts

to justify (one's actions, esp discreditable actions, or beliefs) with plausible reasons, esp after the event

However, a rationalizer's explanations generally have an air of plausibility, even when they are an obvious subterfuge.

Weasel is a common slang term for such a person. Urban Dictionary entries lightly edited for grammar and spelling:

• Shifty, scheming person that will do whatever they need to to escape whatever they fear in the moment.

• A cunning, sneaky person. On internet forums they are often a liar, fake or poser. Will try to extract information from you to share with others and / or use it against you. Uses shiftiness to evade an obligation, commitment or duty. They will make promises then renege on them.

• One who is particularly cunning, conniving, shifty, sneaky, scheming and down right sly. Weasels are known for their ability to adapt to any situation and to manipulate it to suit their own weasel ways.

The term is especially appropriate to describe somebody who twists the truth to get something they want or to evade responsibility. There's also a term for the kind of equivocations that weasels use to twist the truth: weasel words. Many style guides exhort writers to avoid weasel words.