= Solution
The $x$ axis is an <invariant subspace> on which $r$ acts as $+1$ and $s$ as $-1$. The $yz$ plane carries the <irreducible representation> $E$ from <2/b>. Thus
$$
\boxed{R\cong\chi_{-,+}\oplus E,\qquad\chi_R=(3,-1,-1,1,-1).}
$$
The <character of a tensor product> is the product of the <characters>, so $\chi_{R\otimes R}=(9,1,1,1,1)$. Weighted <character inner products> give
$$
\boxed{R\otimes R\cong
2\chi_{+,+}\oplus\chi_{+,-}\oplus\chi_{-,+}\oplus\chi_{-,-}\oplus2E.}
$$
For the trivial <character>, the multiplicity is $(9+1+2+2+2)/8=2$. For any other one-dimensional <character>, the weighted sum over the nonidentity classes is $-1$, giving $(9-1)/8=1$. For $E$, it is $(9\cdot2-2)/8=2$.
Equivalently, $E\otimes E$ is the sum of all four one-dimensional <representations>, and tensoring $E$ by $\chi_{-,+}$ gives the same two-dimensional <irreducible representation>. The <dimensions> in the boxed decomposition sum to nine, as required.
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