= Magnetic reversal and domain averaging
{title2=$\varepsilon^{\rm mag}(-\mathbf m)=\varepsilon^{\rm mag}(\mathbf m)$}
A full $180^\circ$ reversal leaves <saturated isotropic magnetostrictive strain> unchanged. A $90^\circ$ rotation generally changes it. In the simplest unconstrained orientation average over <magnetic domains>, replace $m_i m_j$ by $\langle m_i m_j\rangle$, not by $\langle m_i\rangle\langle m_j\rangle$. Equal populations pointing along $+z$ and $-z$ have zero net <magnetization> but the same preferred <strain> as a single domain along $z$. An isotropic orientation distribution instead has $\langle m_i m_j\rangle=\delta_{ij}/3$ and zero average preferred <strain>. Elastic compatibility between real <magnetic domains> can modify the macroscopic response.
Reversing an applied <magnetic field> does not guarantee that all microscopic orientations retrace the same path. <Hysteresis>, <magnetic domain-wall pinning>, and changes in domain populations can alter the measured curve. The reversal identity concerns the same pair of magnetic states with opposite local <magnetization>; it is not a universal prediction that an arbitrary experimental sweep is a single-valued even function of applied <magnetic field>.
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