The term "rigid pigeon" refers to a state or condition characterized by extreme stiffness, inflexibility, or strict adherence to rules, norms, or procedures. It stands in direct contrast to the notion of being "loosey goosey," implying a lack of relaxation, adaptability, or freedom in approach, behavior, or mindset. This phrase typically describes someone or something that is overly structured, constrained, or resistant to change, often lacking spontaneity or the ability to improvise.
The manager's rigid pigeon approach stifled innovation within the project.