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Understood. But how did it come to be that a rule developed (or even that one could possibly develop) such that for every symmetry in QM there is a corresponding Conservation Law?

First, allow me to correct my earlier response. It turns out that it isn't limited to QM. For each continuous symmetry in nature (wherein the rules of physics are invariant under some transformation) there is a corresponding Conservation Law, and vice-versa.

Spatial translational invariance <=> Conservation of Momentum

Spatial rotational invariance <=> Conservation of Angular Momentum

Temporal invariance <=> Conservation of Energy (and, by extension via Relativity: E=mc²) Matter.

Given the "nature" of nature, logic requires that the symmetries be equivalent to the corresponding Conservation Law. It's based on Noether's Theorem.

A web search will take to sites that can tell you more details that I can.

There are also discrete symmetries in QM that give rise to conservation of various properties.

Hope that helps.

Given the "nature" of nature, logic requires that the symmetries be equivalent to the corresponding Conservation Law.

Thanks for your response. This comment of yours strikes at the heart of the matter.

How did it come to be that nature has the nature it has? How did it come to be that nature is logical in the way it is? How did it come to be that there are symmetries at all?

I don't really expect a scientific answer to these questions since they are not really scientific in nature. The answers get to the underlying reason(s) for how physical laws developed. One can always push the envelop back and state that y is the result of x; for example, that time was created in the Big Bang. But how did a state of affairs come to be so that the creation of time could even be a possible result of the Big Bang?