Lorraine at Scottish Thyroid Petition posted a study which attempts to shed some light on this question. The full PDF is available free. PR
"Adrenal function might be impaired in patients with primary hypothyroidism....In conclusion, patients with different degrees of intensity of primary hypothyroidism had improved cortisol response after reaching euthyroidism. The incidence of adrenal insufficiency was 6.7–18.3% and more than 50% of the cases had a normal cortisol response after L-T4 therapy."