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This paper investigates joint wireless information and energy transfer in a two-user MIMO interference channel, in which each receiver either decodes the incoming information data (information decoding, ID) or harvests the RF energy (energy harvesting, EH) to operate with a potentially perpetual energy supply. In the two-user interference channel, we have four different scenarios according to the receiver mode - (ID1, ID2), (EH1, EH2), (EH1, ID2), and (ID1, EH2). While the maximum information bit rate is unknown and finding the optimal transmission strategy is still open for (ID1, ID2), we have derived the optimal transmission strategy achieving the maximum harvested energy for (EH1, EH2). For (EH1, ID2), and (ID1, EH2), we find a necessary condition of the optimal transmission strategy and, accordingly, identify the achievable rate-energy (R-E) tradeoff region for two transmission strategies that satisfy the necessary condition - maximum energy beamforming (MEB) and minimum leakage beamforming (MLB). Furthermore, a new transmission strategy satisfying the necessary condition - signal-to-leakage-and-energy ratio (SLER) maximization beamforming - is proposed and shown to exhibit a better R-E region than the MEB and the MLB strategies. Finally, we propose a mode scheduling method to switch between (EH1, ID2) and (ID1, EH2) based on the SLER.