Not in the way you suggest: a sequence of letters with the property of being immutable by any type of permutation on its constituent bases. I would be interested if anything of the likes exists in mathematics. 

Having said that, there are examples of sequences that seem to have such property but only in the light of evolution and the biological processes that come with it:

[Ultra-conserved regions](https://en.wikipedia.org/wiki/Ultra-conserved_element) have been described as sequences of about 200 base pairs that appear to be exactly the same from humans to mice. They have not experience change in about ~300 million years of evolution (!) and exposure to the various forces of mutation. 

The reason for this is not an underlying 'invariant to permutation' property, but rather strong negative selection that prevents individuals that acquire changes on the sequence from being able to pass the changes over to their progeny, thus effectively maintaining only the original version in the population.

The exact mechanisms, functions and properties of these regions are under research. You might want to see if the invariant properties you have in mind have anything to do with these interesting regions.