We study the asymptotic behavior of the $ν$-symmetric Riemman sums for functionals of a self-similar centered Gaussian process $X$ with increment exponent $0<α<1$. We prove that, under mild assumptions on the covariance of $X$, the law of the weak $ν$-symmetric Riemman sums converge in the Skorohod topology when $α=(2\ell+1)^{-1}$, where $\ell$ denotes the smallest positive integer satisfying $\int_{0}^{1}x^{2j}ν(dx)=(2j+1)^{-1}$ for all $j=0,\dots, \ell-1$. In the case $α>(2\ell+1)^{-1}$, we prove that the convergence holds in probability.