We address the problem of estimating the harmonics of a two-dimensional exponential mixture. A new algorithms that exploits the multiple-invariance structure is proposed. Unlike previous ESPRIT type methods for two- or multi-dimensional harmonic estimation that are based on joint diagonalization the new algorithm uses polynomial rooting to efficiently obtain the parameters of interest. We reveal a close relations to the recently proposed MD-RARE and show that in the new algorithm, referred to as the rooted multiple-invariance MD-ESPRIT, the degrees of the characteristic polynomials are reduced by a factor two compared to the MD-RARE polynomials. This leads to lower computational complexity and improved numerical stability of the estimation procedure.