Irresponsibility (recklessness) is not the opposite of Responsibility, it is the negative of Responsibility in the same way that a lack of Faith is the negative of Faith while the opposite of Faith is Disbelief.
Practically if you have a responsibility to someone or something it makes it difficult to maintain commitments elsewhere (integrity). Conversely, committing to someone increases one’s sense of responsibility to that same person. The first example shows the dynamic pairs in opposition; the second in collaboration (positive).