Nondeterminism is essential in small two-way finite automata with few reversals

On every n-long input, every two-way finite automaton (2fa) can reverse its input head O(n)O(n) times before halting. A 2fawith few reversals   is an automaton where this number is only o(n)o(n). For every h, we exhibit a language that can be recognized by an h-state nondeterministic 2fa with few reversals, but requires Ω(2h)Ω(2h) states on every deterministic 2fa with few reversals.