Emergency department triage assigns patients an acuity score that determines treatment priority, and clinical evidence documents persistent gender disparities in human acuity assessment. As hospitals pilot large language models (LLMs) as triage decision support, a critical question is whether these models reproduce or mitigate known biases. We present EQUITRIAGE, a fairness audit of LLM-based ESI assignment evaluating five models (Gemini-3-Flash, Nemotron-3-Super, DeepSeek-V3.1, Mistral-Small-3.2, GPT-4.1-Nano) across 374,275 evaluations on 18,714 MIMIC-IV-ED vignettes under four prompt strategies. Of 9,368 originals, 9,346 are paired with a gender-swapped counterfactual. All five models produced flip rates above a pre-registered 5% threshold (9.9% to 43.8%). Two showed directional female undertriage (DeepSeek F/M 2.15:1, Gemini 1.34:1); two were near-parity; one had high sensitivity with weak male-direction asymmetry. DeepSeek's directional bias coexisted with a low outcome-linked calibration gap (0.013 against MIMIC-IV admission), a Chouldechova-style dissociation between within-group calibration and between-pair counterfactual invariance. Demographic blinding reduced Gemini's flip rate to 0.5%; an age-preserving blind variant left DeepSeek with residual F/M 1.25, implicating age as a residual channel. Chain-of-thought prompting degraded accuracy for all five models. A two-model ablation reveals opposite underlying mechanisms for the same directional phenotype: in Gemini the signal is emergent in the combined name+gender swap, while in DeepSeek the gender token alone carries it. EQUITRIAGE shows that group parity, counterfactual invariance, and gender calibration are distinct fairness properties, that intervention effectiveness is model-dependent, and that per-model counterfactual auditing should precede clinical deployment.