Climbing birds mostly use a tripod: two feet on the bark and a stiffened tail pressed against it as a third support. That arrangement works going up and fails going down. Nuthatches abandoned the tail brace entirely; their tails are short and soft, and the family has no such adaptation.
Instead a descending nuthatch places one foot high and behind, hooked into bark, while the lower foot takes the weight, so the bird hangs from the upper leg rather than pressing against the trunk. The posture gives a different view into crevices. Studies of foraging in mixed-species flocks find that nuthatches and upward-climbing species take prey from partly different microsites on the same trunk, so the head-down habit is a genuine access to overlooked food.
The name records the other habit. A nuthatch wedges a nut or large seed into a bark fissure and hammers it apart with the bill, an act older English called hacking or hatching the nut. It also caches seeds under bark flakes and recovers them in winter.