Large Language Models (LLMs) are increasingly used to annotate learning interactions, yet concerns about reliability limit their utility. We test whether verification-oriented orchestration-prompting models to check their own labels (self-verification) or audit one another (cross-verification)-improves qualitative coding of tutoring discourse. Using transcripts from 30 one-to-one math sessions, we compare three production LLMs (GPT, Claude, Gemini) under three conditions: unverified annotation, self-verification, and cross-verification across all orchestration configurations. Outputs are benchmarked against a blinded, disagreement-focused human adjudication using Cohen's kappa. Overall, orchestration yields a 58 percent improvement in kappa. Self-verification nearly doubles agreement relative to unverified baselines, with the largest gains for challenging tutor moves. Cross-verification achieves a 37 percent improvement on average, with pair- and construct-dependent effects: some verifier-annotator pairs exceed self-verification, while others reduce alignment, reflecting differences in verifier strictness. We contribute: (1) a flexible orchestration framework instantiating control, self-, and cross-verification; (2) an empirical comparison across frontier LLMs on authentic tutoring data with blinded human "gold" labels; and (3) a concise notation, verifier(annotator) (e.g., Gemini(GPT) or Claude(Claude)), to standardize reporting and make directional effects explicit for replication. Results position verification as a principled design lever for reliable, scalable LLM-assisted annotation in Learning Analytics.