= Solution
The high-Reynolds-number balance gives positive mean <vortex stretching>. The <Betchov relation> therefore gives $\langle abc\rangle<0$. To connect this to a fixed longitudinal velocity derivative, additionally assume <isotropic turbulence>, or average over the direction of that derivative. For a unit vector $\mathbf n$, the derivative is $d=\mathbf n^TS\mathbf n$. Uniform directional averaging, with $\operatorname{tr}S=0$, gives
$$
\overline{d^2}=\frac2{15}\operatorname{tr}S^2,\qquad
\overline{d^3}=\frac8{105}\operatorname{tr}S^3=\frac8{35}abc.
$$
For example these identities follow by averaging products of direction cosines on a sphere. Hence in <isotropic turbulence>
$$
\langle(\partial_xu_x)^3\rangle=\frac8{35}\langle abc\rangle
=-\frac2{35}\langle\omega_i\omega_jS_{ij}\rangle<0,
$$
and division by the positive derivative <variance> to the power $3/2$ gives \b[negative <longitudinal velocity-gradient skewness>]. Homogeneity alone does not fix the orientation statistics and does not establish this sign for every chosen coordinate direction.
Order the <principal strain rates> as $a\ge b\ge c$. Except at degeneracies, $a>0$ and $c<0$, so $abc<0$ corresponds to $b>0$: two extensional directions and one compressive direction, or <bi-axial strain>. The negative average of $abc$ says that such configurations dominate this cubic amplitude-weighted statistic over axial-extension configurations with $b<0$. In the usual physical interpretation, comparably intense strain events are therefore biased toward bi-axial extension.
An unweighted frequency claim needs more information. As a counterexample to deducing it solely from the cubic average, assign strain <eigenvalues> $(1,1,-2)$ to 10% of samples and $(-0.1,-0.1,0.2)$ to 90%, with randomly oriented <eigenvectors>. Then $\langle abc\rangle=-0.1982<0$, despite bi-axial extension occurring in only 10% of the samples. This example addresses what the <moment> inequality can infer, not a construction of a complete turbulent <velocity field>. A measured frequency preference is a further statistical observation.
The geometric significance is that a fluid element stretches in two directions and contracts in the third, becoming sheet-like while preserving volume. Distributed <vorticity> is swept into thin layers and its component along an extensional direction is amplified. The associated <shear layers> can subsequently roll up and produce <vortex tubes>. Thus the weighted strain bias helps explain sheet formation in the <energy cascade> even though the most intense dissipative structures can be tubes.
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