Existing Large Language Model (LLM) serving systems prioritize maximum throughput. They often neglect Service Level Objectives (SLOs) such as Time to First Token (TTFT) and Time Per Output Token (TPOT), which leads to suboptimal SLO attainment. This paper introduces SCORPIO, an SLO-oriented LLM serving system designed to maximize system goodput and SLO attainment for workloads with heterogeneous SLOs. Our core insight is to exploit SLO heterogeneity for adaptive scheduling across admission control, queue management, and batch selection. SCORPIO features a TTFT Guard, which employs least-deadline-first reordering and rejects unattainable requests, and a TPOT Guard, which utilizes a VBS-based admission control and a novel credit-based batching mechanism. Both guards are supported by a predictive module. Evaluations demonstrate that SCORPIO improves system goodput by up to 14.4X and SLO adherence by up to 46.5% compared to state-of-the-art baselines.