If South-East becomes West, North-East becomes South and so on, then what will East become?

Let's trace the rotation angle of the standard directions:
* Standard South-East ($\text{SE}$) shifts to West ($\text{W}$). On a standard direction compass, moving from $\text{SE}$ clockwise to $\text{W}$ is an angle of $135^{\circ }$ clockwise.
* Standard North-East ($\text{NE}$) shifts to South ($\text{S}$). Moving from $\text{NE}$ clockwise to $\text{S}$ is also a $135^{\circ }$ clockwise rotation.

Therefore, every direction shifts by $135^{\circ }$ in the clockwise direction.
To find what East ($\text{E}$) becomes, we rotate East by $135^{\circ }$ clockwise:

* $90^{\circ }$ clockwise from East is South.
* Another $45^{\circ }$ clockwise from South is South-West.

Thus, East will become South-West.