That doesn't strike me so much as an assumption as it is just a statement that it's always going to be possible to describe the world in terms of some
symmetries. One can show, for instance, that it is possible to take purely random, time-variant laws of nature, and show that due to the ambiguity of the time coordinate, there exists a time-invariant way of describing the system:
I tend to expect that in a sense, then, the existence of some
symmetries is an inevitability, just based upon how we approach understanding the world.
That said, physical theories don't just rest at the most basic of symmetries like time invariance. Hypothetical high-energy laws of physics typically consider all of physical law as stemming from some fundamental construct (particles, strings, what have you) that obey some very specific symmetries.
I don't think the most basic of symmetries are really an assumption so much as order we impose on the world by the way in which we describe it.