If the metric is a (0,2)-tensor then it transforms twice covariently under a basis transformation, but in special relativity the metric seems to be defined globally as: $$\eta_{\mu\nu}=\begin{pmatrix}-1 && 0 && 0 && 0 \\ 0 && 1 && 0 && 0 \\ 0 && 0 &&1&&0\\0&&0&&0&&1\end{pmatrix}$$ in most texts...