Lower-than-random ROC If I have an ROC for a single classifier [y(x) in the range 0...1] that is 'worse than random', namely the AUC of the ROC is less than 0.5,  would a classifier that reversed the class predictions [y'(x)=1-y(x)] be better than random to the same degree?
$y'(x)$ means you will work with $ROC' = ROC^{-1}$ (inverse of $ROC$), as all true positive will be falsely negative and vice versa. Therefore, $AUC' = 1 - AUC$ (As ROC is an increasing function and inside a unit square), and your answer is yes.