As Danny Rorabaugh's OEIS sequence A262312 suggests, the number of binary words of length $n$ that begin with a even-length palindromic prefix is the same as the number of binary words of length $n$ that have a matching prefix and suffix (sometimes called a "bifix").
I'm interested in finding a bijection $$ \phi_n\colon \big\{s \in \{0,1\}^n : s \text{ has an even-length palindromic prefix}\big\} \rightarrow \big\{s \in \{0,1\}^n : s \text{ has a bifix}\big\} $$ between these sets such that the length of the longest palindromic prefix of $s$ is equal to the total length of the longest bifix of $\phi_n(s)$.
For instance, a map $\underline{1001}1 \mapsto \underline{01}1\underline{01}$ is good, because the palindromic prefix on the left has length four, which is equal to the (total) length of the bifix on the right. However, the map $\underline{00}010 \mapsto \underline{00}1\underline{00}$ is not acceptable, because the palindromic prefix on the left has length two, but the (total) length of the bifix on the right has length four.
Is there a natural way to create a length-preserving bijection like this?
