This paper investigates energy-efficient inter-satellite communication in Low Earth Orbit (LEO) networks, where satellites exchange both buffered and newly generated data through half-duplex inter-satellite links (ISLs). Due to orbital motion and interference-prone directional asymmetry, the achievable ISL capacities in opposite directions vary dynamically, leading to inefficient utilization under conventional fixed or alternating duplex modes. To address this, we propose a Flexible Duplex (FlexD) scheme that adaptively selects the ISL transmission direction in each slot to maximize instantaneous end-to-end sky-to-ground throughput, jointly accounting for ISL quality, downlink conditions, and queue backlogs. A unified analytical framework is developed that transforms the bottleneck rate structure into an equivalent SINR domain, enabling closed-form derivations of throughput outage probability and energy efficiency under deterministic ISLs and Rician satellite-to-ground fading. The analysis reveals distinct operating regions governed by ISL and backlog constraints and provides tractable bounds for ergodic rate and energy efficiency. Numerical results confirm that FlexD achieves higher reliability and up to 30% improvement in energy efficiency compared with conventional half- and full-duplex schemes under realistic inter-satellite interference conditions.