Given a string $T$ on an alphabet of size $σ$, we describe a bidirectional Burrows-Wheeler index that takes $O(|T|\logσ)$ bits of space, and that supports the addition \emph{and removal} of one character, on the left or right side of any substring of $T$, in constant time. Previously known data structures that used the same space allowed constant-time addition to any substring of $T$, but they could support removal only from specific substrings of $T$. We also describe an index that supports bidirectional addition and removal in $O(\log{\log{|T|}})$ time, and that occupies a number of words proportional to the number of left and right extensions of the maximal repeats of $T$. We use such fully-functional indexes to implement bidirectional, frequency-aware, variable-order de Bruijn graphs in small space, with no upper bound on their order, and supporting natural criteria for increasing and decreasing the order during traversal.