  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.
