Wireless power transfer (WPT) technique has evolved into a revolutionary technology for realizing green and self-sustainable wireless networks. Meanwhile, the newly emerging intelligent reflecting surface (IRS) composed of massive low-cost and low-power reflecting elements has also been anticipated to greatly enhance both spectral and energy efficiencies of wireless communications. In this paper, we study throughput maximization in the IRS-aided multiple-input multiple-output (MIMO) wireless powered communication networks (WPCNs) by jointly optimizing the energy and information covariance matrices, the downlink/uplink (DL/UL) IRS reflect beamforming vectors and the DL/UL time allocation. Under the assumption of separate DL/UL IRS reflection pattern, we firstly study the multi-user scenario and develop a 3-block alternating optimization algorithm for finding a high-quality suboptimal solution. To draw more insightful conclusions, the single-user scenario is then studied, in which two low-complexity alternative schemes in the asymptotically low-SNR and high-SNR regimes are proposed, respectively. Furthermore, for the common DL/UL IRS reflection pattern with the intensive coupling between DL energy transfer and UL information transmission, we consider a two-stage scheme for maximizing UL throughput in the multi-user scenario, while the proposed low-complexity schemes for the single-user scenario are still applicable after some modifications. Numerical results are carried out to illustrate the superior UL throughput performance of our proposed algorithms over benchmarks. It is also revealed that the well-known doubly near-far problem in MIMO WPCNs can be effectively overcame by properly deploying the IRS.