Will Sawin's answer necessitates $n-1$ passes over the polynomial $P$, checking for equality. Ryan's comment to Will's answer brings that down to $2$ passes, but with (somewhat) expensive operations to be done.
You can do it in a single pass with only cheap operations, assuming your polynomial $P$ is given in expanded form in the monomial basis. First, you know that, if it is symmetric, it can be rewritten as a polynomial in the symmetric polynomials. So, march through all the coefficients of $P$, figuring out the 'signature' of each monomial (i.e. set of degrees) you encounter; then make sure that the coefficient for each signature is constant, and that you encounter enough such monomials for each degree. This a linear pass on $P$, and storage $O(m)$ where $m$ is the number of different symmetric polynomials which actually occur in $P$.
Of course, if your polynomial is not presented in expanded form in the monomial basis, the above will not work.