It is proved that a two-way alternating finite automaton (2AFA) with n states can be transformed to an equivalent one-way nondeterministic finite automaton (1NFA) with f(n)=2^{Θ(n logn)} states, and that this number of states is necessary in the worst case already for the transformation of a two-way automaton with universal nondeterminism (2Π₁FA) to a 1NFA. At the same time, an n-state 2AFA is transformed to a 1NFA with (2 ⁿ -1)²+1 states recognizing the complement of the original language, and this number of states is again necessary in the worst case. The difference between these two trade-offs is used to show that complementing a 2AFA requires at least Ω(n logn) states.