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.