When Landau (Classical Theory of Fields) introduces the Levi-Civita symbol, he writes
With respect to rotations of the coordinate system, the quantities $\epsilon^{iklm}$ behave like the components of a tensor; but if we change the sign of one or three of the coordinates the components of that tensor, being defined as the same in all coordinate systems, do not change, whereas some of the components of a tensor should change sign.
I am completely Ok with understanding that $\epsilon^{iklm}$ is a pseudotensor. What confuses me in this quote is the one/three choice, implying that if I reflect two coordinates none of the components of a (real) tensor should change sign. But as far as I can see, some of the components of a tensor will change sign whenever I reflect any number of coordinates. For example, if I have a two-rank tensor $A^{ij}$ and reflect two coordinates, say, $x'^0 = x^0$, $x'^1=-x^1$, $x'^2=-x^2$, $x'^3=x^3$, then, using the transformation rule for tensors, I get that, e.g., $A'^{01}=-A^{01}$, so some components of a tensor do change sign if I reflect two axes. If I am correct, why does Landau single out odd number of reflections?