In this paper, a solar-powered bidirectional communication system is studied for a pair of energy harvesting (EH) nodes that intend to communicate with each other over wireless fading channels. The conventional time-division duplex (TDD) transmission is revisited by proposing a stochastic resource scheduling scheme to minimize an average rate outage probability based on a Markov decision process (MDP) design framework. Different from the conventional TDD transmission, the proposed scheme can adjust the link direction and energy expenditure for data transmissions between the two EH nodes, in response to the dynamics of the solar EH, channel fading and battery storage conditions. A downstairs threshold structure is theoretically proved under a special optimal on-off policy, in which two-dimensional thresholds pinpoint the interplay between the transmission actions and the available energy in the batteries of the two nodes. Also the optimal on-off policy at asymptotically high signal-to-noise power ratios (SNRs) is revealed. The outage performance of the proposed stochastic resource scheduling scheme is validated by extensive computer simulations, and it shows that the proposed optimal MDP policy can achieve significant performance gains over the combinations of other compared schemes, including round-robin and battery state-oriented link scheduling schemes, and greedy and conservative energy scheduling schemes.