No, the electrons actually move in the same direction always, just as a free falling object always moves towards the locally dominant mass. We could call the direction of electron movement "right".
It's the magnetic field that has the arbitrary sign convention. You can't determine the direction of a magnetic field from observations without using the right hand rule.
You don't have to know the sign, you could tell which way the electrons are going.
We tell the aliens: you take all sorts of organics with liquid inside (on earth we typically use a lemon, but other organics will do), and put a stake of element 29 in one side and a stake of element 30 in the other. You then connect them with a length of element 29. Keep trying different organics until you find one that works. They might not have lemons, but there is a good chance that something will eventually work.
You'd need to predefine what the atomic numbers mean and some other things, but we're already assuming some level of communication already being established so this aspect is not far-fetched.
Now arrange the apparatus such that the element 29 side is near you, and the element 30 side is far from you. Additionally, ensure that the wire is up (further from the locally dominant gravitational mass). Now place an election between the "lemon" and the wire. See which way the election moved? That was right (or left, I don't remember).
Set up whatever apparatus you like, you can always set up its mirror image, and then the electrons are going in the opposite direction. So which is right and which is left?
Remember you haven't yet established with the alien what you mean by right and left, so you won't be able to instruct it to set up your apparatus rather than its mirror image, and of course you can't appeal to the right-hand rule.
This problem has no solution because electromagnetism is right-left symmetric. You would need to use the weak decay (and assume that your alien is made of matter rather than antimatter)
I addressed that. We know which way the electrons flow from the anode to the cathode. I specifically chose this example because of the right-hand rule.