Bijective counting of involutive Baxter permutations

We enumerate bijectively the family of involutive Baxter permutations according to various parameters; in particular we obtain an elementary proof that the number of involutive Baxter permutations of size $2n$ with no fixed points is $\frac{3\cdot 2^{n-1}}{(n+1)(n+2)}\binom{2n}{n}$, a formula originally discovered by M. Bousquet-M\'elou using generating functions. The same coefficient also enumerates planar maps with $n$ edges, endowed with an acyclic orientation having a unique source, and such that the source and sinks are all incident to the outer face.