Set the Transform Orientations to Local, switch to the Move tool, switch the Orientation to Default and you'll see that the origin orientation is not what you think it is. To correct that you could for example align the eyelash origin with the head orientation. Also, are you sure that you want to animate such a high-poly head?
To align the eyelash origin with the head origin, select the head and create a custom orientation:
Enable the Affect Only Origins option:
Select the eyelash and Object > Transform > Align to Transform Orientation:
Disable Affect Only Origins:
Also don't forget to switch back to Global or Local Transform Orientations:
Do the SX-1 operation on the eyelash, it should work.