We propose a two-sided interconnection-based method for model order reduction of quadratic-bilinear systems. Exploiting the solution of an infinite-dimensional system of Sylvester-like equations we define the “swapped” nonlinear moment by a power series representation. We then consider together the “direct” moment and the “swapped” moment and propose two families of reduced-order models that achieve two-sided approximate moment matching. We illustrate the results of the paper on the problem of reducing the 1-D Burgers’ equation.