Why only orientation-preserving transformations are considered when integrating forms?

Question

The volume form is known to be invariant under a change of coordinates $T$ with $\det(T)>0$, so consequently integral of forms are also invariant. But what happens when the change of coordinates has $\det(T)<0$? Is the integral invariant under transformations with negative determinant? In Calculus of one variable, when the determinant is negative one would just reverse the limit of integration, why we don't do the same when dealing with forms? In Calculus of multivariable, the absolute value of the determinant is taken, so I guess my question is related to why there's an absolute value there (is it put by hand or does it show up naturally?).