  We construct a new family of random permutons, called skew Brownian permuton,
which describes the limits of several models of random constrained
permutations. This family is parametrized by two real parameters.
  For a specific choice of the parameters, the skew Brownian permuton coincides
with the Baxter permuton, i.e., the permuton limit of Baxter permutations. We
prove that for another specific choice of the parameters, the skew Brownian
permuton coincides with the biased Brownian separable permuton, a one-parameter
family of permutons previously studied in the literature as the limit of
uniform permutations in substitution-closed classes. This brings two different
limiting objects under the same roof, identifying a new larger universality
class.
  The skew Brownian permuton is constructed in terms of flows of solutions of
certain stochastic differential equations (SDEs) driven by two-dimensional
correlated Brownian excursions in the non-negative quadrant. We call these SDEs
skew perturbed Tanaka equations because they are a mixture of the perturbed
Tanaka equations and the equations encoding skew Brownian motions. We prove
existence and uniqueness of (strong) solutions for these new SDEs.
  In addition, we show that some natural permutons arising from Liouville
quantum gravity spheres decorated with two Schramm-Loewner evolution curves are
skew Brownian permutons and such permutons cover almost the whole range of
possible parameters. Some connections between constrained permutations and
decorated planar maps have been investigated in the literature at the discrete
level; this paper establishes this connection directly at the continuum level.
Proving the latter result, we also give an SDE interpretation of some
quantities related to SLE-decorated Liouville quantum gravity spheres.
