Necessary and sufficient analytical conditions are given for homogeneous multiple Wiener-Itô integral processes (MWIs) to be mixing, and sufficient conditions are given for mixing of general square-integrable Gaussian-subordinated processes. It is shown that every finite or infinite sum Y of MWIs (i.e. every real square-integrable stationary polynomial form in the variables of an underlying weakly mixing Gaussian process) is mixing if the process defined separately by each homogeneous-order term is mixing, and that this condition is necessary for a large class of Gaussian-subordinated processes. Moreover, for homogeneous MWIs Y1, for sums of MWIs of order ≤ 3, and for a large class of square-integrable infinite sums Y1, of MWIs, mixing holds if and only if Y2 has correlation-function decaying to zero for large lags. Several examples of the criteria for mixing are given, including a second-order homogeneous MWI, i.e. a degree two polynomial form, orthogonal to all linear forms, which has auto-correlations tending to zero for large lags but is not mixing.