  In any geometrically nonlinear quadratic Cosserat-micropolar extended
continuum model formulated in the deformation gradient field $F :=
\nabla\varphi: \Omega \to \mathrm{GL}^+(n)$ and the microrotation field $R:
\Omega \to \mathrm{SO}(n)$, the shear-stretch energy is necessarily of the form
\begin{equation*} W_{\mu,\mu_c}(R\,;F) := \mu\,\left\lVert{\mathrm{sym}(R^T F -
\boldsymbol{1})}\right\rVert^2 + \mu_c\,\left\lVert{\mathrm{skew}(R^T F -
\boldsymbol{1})}\right\rVert^2\;, \end{equation*} where $\mu > 0$ is the Lam\'e
shear modulus and $\mu_c \geq 0$ is the Cosserat couple modulus. In the present
contribution, we work towards explicit characterizations of the set of optimal
Cosserat microrotations
$\mathrm{argmin}_{R\,\in\,\mathrm{SO}(n)}{W_{\mu,\mu_c}(R\,;F)}$ as a function
of $F \in \mathrm{GL}^+(n)$ and weights $\mu > 0$ and $\mu_c \geq 0$. For $n
\geq 2$, we prove a parameter reduction lemma which reduces the optimality
problem to two limit cases: $(\mu, \mu_c) = (1,1)$ and $(\mu,\mu_c) = (1,0)$.
In contrast to Grioli's theorem, we derive non-classical minimizers for the
parameter range $\mu > \mu_c \geq 0$ in dimension $n\!=\!2$. Currently,
optimality results for $n \geq 3$ are out of reach for us, but we contribute
explicit representations for $n\!=\!2$ which we name
$\mathrm{rpolar}^{\pm}_{\mu,\mu_c}(F) \in \mathrm{SO}(2)$ and which arise for
$n\!=\!3$ by fixing the rotation axis a priori. Further, we compute the
associated reduced energy levels and study the non-classical optimal Cosserat
rotations $\mathrm{rpolar}^\pm_{\mu,\mu_c}(F_\gamma)$ for simple planar shear.
