The torso is the part from the hips up to the neck (approximately) but doesn't include the head or the arms so it wouldn't refer to the entire upper part of the body. I can't think of any suitable terms, like teechar.
Also, you said that "side" means the right/left half of a body, but that isn't strictly true. If someone says "I have a pain in my side", it refers only to the left or right side of their torso, probably somewhere around the ribs. It wouldn't refer to the side of their leg or arm or head, etc.