Mr. Gutwrench is a made-up teacher that Leslie imagines her teacher Mr. Garrison to be like.
Mr. Gutwrench is described by Leslie as "the meanest teacher alive." However, he turns out to be a doppelganger for the kind and mild-mannered Mr. Garrison.
When the principal puts Leslie in his class in #43: “Back to School”, Mr. Gutwrench immediately makes fun of her torn dress, gives her a 10,000-word essay to write and yells at her for hanging out with her friends during the summer. Then, after Leslie makes what she called an "innocent little statement," he gives out homework.
In #44: “It Sure Seems Like It to Me”, his antics continue. However, in the same episode, it's established that his actual name is Mr. Garrison, and that his personality is the opposite of how Leslie was describing him — thus proving she was blowing him out of proportion.
Any family he may have is unknown.