Symmetrizing Polynomial

Question

For a polynomial $f(x_1, \dots, x_n)$, we define the operator $T_{\theta, i,j}$ which takes $f$ to $\theta f(x_1, \dots, x_{i-1}, x_j, \dots, x_{j-1}, x_i, \dots) + (1-\theta) f(x_1, \dots, x_n)$. In words, this is the expected polynomial when one swaps indices $i$ and $j$ with probability $\theta$. Does there exist a finite sequence of $T_{\theta_k, i_k, j_k}$ such that for any polynomial, the outcome of applying this sequence of operators is the symmetrized polynomial (i.e. the expected polynomial over a uniform distribution of all permutations of the indices)?