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.]

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