In recent years, there have been notable advancements in the theoretical research of existing meta-heuristic algorithms and in the development of novel algorithms. However, the research on the definition and construction methods of the so-called “mutation” operator, a concept widely used in existing meta-heuristic algorithms, remains notably insufficient. In addition, how to directly construct a meta-heuristic algorithm under the guidance of stability and invariance analysis continues to pose a challenge that has yet to resolve. This paper carefully investigates and rigorously defines the concepts of “mutation operator” and “mutation-driven meta-heuristic algorithm” as found in the literature. According to these definitions, a novel meta-heuristic algorithm, called Multi-direction guided Mutation-driven Stable swarm Intelligence algorithm with Translation and Rotation invariance (M2SITRI) is proposed, and the corresponding mutation operator is carefully designed based on the analysis of translation invariance, rotation invariance, order-1 stability, and order-2 stability. The stability and invariance properties of the M2SITRI algorithm are validated through numerical simulations, and its superior solving performance is demonstrated by benchmarking it against several other well-known existing meta-heuristic algorithms across a variety of test problems.