Misrepresent, falsify, belie, garble mean to present or represent something in a manner that is contrary to the truth.
Misrepresent usually implies an intent to represent or portray falsely, and therefore may take as its subject not only a person or his utterance but also such things as an organization, a political platform, or a cartoon; it often carries a suggestion of deliberate lying, but it may suggest bias or prejudice or a desire to do injustice.
Falsify implies a perversion of the truth, either by deliberately altering the facts in whole or in part or by giving something an appearance that does not accord with the truth or reality.
Belie implies an impression given that contradicts or is at variance with the truth or the facts; it usually takes as its subject an appearance, a manner, or a form of speech, and lacks the implication of intent that is usually present in the preceding words.
Garble implies a mutilation or distortion of statements, testimony, evidence, or messages that need not be deliberate but that creates a wrong impression of the original and frequently gravely alters its tone or implications.