Taking a rectangular coil, in a constant magnetic field, it is understandable that BANcos(theta) should apply for the magnitude of the flux linkage. However if the coil is turned through 180 degrees why should the flux linkage then become negative, implying that the emf induced is also negative? As flux linkage is defined in terms of the area swept out, I do not see how this could come about beig negative as the ending position seems to be equivelent to the starting one in terms of B, A and N.
If you consider a constant current flowing in your coil as you rotate it, you will see it at first rotating clockwise, then, after the 180 degree flip, anticlockwise.
If you create the same change in flux, but flip the coil, then you will induce a current in the opposite direction in the frame of the coil. In other words, while it will still look "clockwise" if you look at the coil from a distance, if you had painted little arrows on the coil you would see that in one instance the current flows in the direction of the arrows, and in the other instance flows against it.