In this paper, the Pareto-optimal beam structure for multi-user multiple-input multiple-output (MIMO) interference channels is investigated and a necessary condition for any Pareto-optimal transmit signal covariance matrix is presented for the K-pair Gaussian MIMO interference channel. It is shown that any Pareto-optimal transmit signal covariance matrix at a transmitter should have its column space contained in the union of the signal spaces of the channel matrices from the transmitter to all receivers. Based on this necessary condition, an efficient parameterization for the beam search space is proposed. The proposed parameterization is given by the product manifold of a Stiefel manifold and a subset of a hyperplane and enables us to construct an efficient beam design algorithm by exploiting its rich geometrical structure and existing tools for optimization on Stiefel manifolds. Reduction in the beam search space dimension and computational complexity by the proposed parameterization and the proposed beam design approach is significant when the number of transmit antennas is much larger than the sum of the numbers of receive antennas, as in upcoming cellular networks adopting massive MIMO technologies. Numerical results validate the proposed parameterization and the proposed cooperative beam design method based on the proposed parameterization for MIMO interference channels.