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The various ingredients of Lorenz theory of available potential energy (APE) are shown to hold the key for understanding how to develop a first-principles theory of lateral stirring and lateral stirring surfaces in the oceans embedded in the study of the full Navier-Stokes equations for compressible seawater. This theory establishes that it is the existence of thermobaric forces acting along isopycnal surfaces that makes stirring in seawater fundamentally different from that in a simple fluid and is the ultimate cause for the non-existence of neutral surfaces. It also establishes that the `true' neutral directions are those perpendicular to an APE-based form of the P vector previously identified by Nycander, contrary to what has been assumed so far. Where thermobaric forces are small enough to be neglected, our theory establishes that the Lorenz reference density (LRD) surfaces entering APE theory are very accurately neutral and represent the relevant definition of lateral stirring surfaces. Where thermobaric forces are large, however, lateral stirring becomes strongly coupled with vertical stirring and complicates the identification of the `right' lateral mixing surfaces. Importantly, rewriting the momentum balance equations in their thermodynamic form using Crocco-Vazsonyi theorem and removing the dynamically inert part of the Bernoulli function proves decisive for obtaining most results. The new results have important implications for the theory of isopycnal analysis and ocean mixing parameterisations.