Builders or engineers will use the term integrity in a different sense. This notion of structural integrity is one we'd like to use in developing our understanding of what Jesus is getting at with his teaching here. Structural integrity suggests that, be it a building or a bridge, or a chest of drawers, the structure as a whole hangs together. Nothing is out of place.
Nothing is superfluous or useless.
As long as each of the parts is in place, the whole structure has strength and stability. It was no accident that Jesus, experienced as a carpenter, contrasted a house built on sand with one built on a solid foundation. The foundation of the house doesn't show. Jesus too was thinking of structural integrity as an illustration of character integrity. When this person is tested is challenged to speak the truth at some cost, or is offered an opportunity to make a quick buck by compromising the truth then cracks can develop quickly.