= Solution
Apply the linear transformation of the <scalar fields> to either time- or space-derivative product. Its coefficients become $\alpha\gamma$ multiplying the square of the first derivative, $\beta\delta$ multiplying the square of the second, and $\alpha\delta+\beta\gamma$ multiplying their cross product. Invariance for arbitrary fields therefore requires
$$
\alpha\gamma=0,\qquad\beta\delta=0,\qquad\alpha\delta+\beta\gamma=1.
$$
The complete real solution is
$$
\boxed{\begin{pmatrix}\alpha&\beta\\\gamma&\delta\end{pmatrix}=\begin{pmatrix}r&0\\0&r^{-1}\end{pmatrix}\quad\text{or}\quad\begin{pmatrix}0&r\\r^{-1}&0\end{pmatrix},\qquad r\in\mathbb R\setminus\{0\}.}
$$
There are four <connected components>, distinguished by the diagonal/antidiagonal choice and the sign of $r$. Only the positive diagonal family is in the <identity component>; write $r=e^s$ to obtain $\phi_1\mapsto e^s\phi_1$, $\phi_2\mapsto e^{-s}\phi_2$. This is the <reciprocal rescaling symmetry of two cross-coupled scalar fields>. The antidiagonal transformations exchange the fields as well as rescale them.
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