Indeed the "symmetrized square-free monomials" seem to generate. 
(Order lexicographically and look what the highest term in a product looks like. 
Now use that to concoct rewriting rules.) 

[Oops! This is less obvious than it seemed. The symmetrizations are 
not with respect to the full symmetric group. In fact it fails for the cyclic 
group of order four, where the square-free case is not enough to generate
all invariants of degree three.]

They also seem to be independent, as the transcendence degree matches.

[Oops! Also wrong. It would contradict the Chevalley–Shephard–Todd theorem.
There may be many orbits of our cyclic group in the set
of square free monomials of a given degree.]

One may wish to check the degree of the full ring as a module over the 
predicted subring.
For example, K[x,y] as a module over K[x+y,xy] has basis 1, x, but why?

[Because of the minimal polynomial (T-x)(T-y) over that subring.
But this reasoning is less helpful for larger degree. 
Nevertheless one may wish to look at our full ring as a (free) module over the
polynomial ring in the elementary symmetric functions.
Is there a basis of that module that is permuted by our cyclic group?
And one really wants the ring structure, not just the vector space.]

Wilberd