No, it does not. The determinant tells you whether two bases have the same or different handedness, not which one is "left" or "right".

It's 2 cosets, one is arbitrarily left handed and the other arbitrarily right handed. If you are in an orientable space :) if not, then there's no global concept of left or right. a

Formulated differently, you cannot determine left- and right-handedness but you can determine same-handedness.