How do you express the covariant cross product? If the covariant cross product is given by $\mathbf{AxB}= \varepsilon^{ijk}A_{j}B_{k}$, then the Levi-Civita tensor must transform contravariantly for the indices to contract. But according to this physicsforums thread, it obeys the transformation law $\epsilon^{i'j'k'}=|\det (\frac{\partial x'}{\partial x})|\varepsilon^{ijk}$, which means that it transforms neither covariantly nor contravariantly. What gives?

The cross product is not a vector but instead a pseudovector. Furthermore the Levi-Civita "tensor" $\varepsilon$ is not a tensor but instead a tensor density.