  Two words $p$ and $q$ are avoided by the same number of length-$n$ words, for
all $n$, precisely when $p$ and $q$ have the same set of border lengths.
Previous proofs of this theorem use generating functions but do not provide an
explicit bijection. We give a bijective proof for all pairs $p, q$ that have
the same set of proper borders, establishing a natural bijection from the set
of words avoiding $p$ to the set of words avoiding $q$.
