No. Heck you can’t even assume that conversing with other humans today. I might be using Google translate into one of the R-L or other direction reading languages. Assuming a screen direction is just a proxy for a shared physical reference. No different to handing me a piece of paper with the directions written on it, just more convoluted. Which is of course why the OP disallows it. Same for commonly known symbols or pictures of known objects.
Only the concepts are necessary, not the symbols.
If you define a space, you can define left or right within that space.
~Max
I didn’t think that would work. 
~Max
You can. But you can’t define it in an unambiguous manner. All the mathematics works either way. You need to apply a convention. By convention we use the right hand rule to define the chirality of our mathematics. But there is no intrinsic reason that we do it this way. You could meet someone from a different culture that has developed the same mathematics in parallel, and have a 50/50 chance they defined it the same way around. All your mathematics would be the same, until you tried to draw something.
I’m a little lost here, can you explain how “greater than” can somehow establish left or right? You can certainly define left or right within a space, but without a common reference object to look at or a known reading order (right-to-left or left-to-right), there’s no way to guarantee that the alien’s world isn’t completely flipped.
You could, for example, say to the alien: “In a 3x3 grid (rows 1-2-3, columns A-B-C) , ‘left’ is defined as being in cell A2, and ‘right’ is in C2”. The alien would dutifully mark cell A2 as being “left” and C2 as “right”. With that established, you could expand the grid size and transmit any image from Earth to the alien (right handed humans, a globe with west marked on the left, etc) using a similar fashion. But you have no idea if the alien reads left-to-right or right-to-left. So the initial grid with columns “A-B-C” may have been marked as “C-B-A” by the alien, and so all of your images are backwards on the alien’s end.
If you asked the alien, they’d say they read left-to-right, just like you do, except they may not in actuality. And there’s no way to tell, short of the cobalt-60 experiment referenced above, or being able to reference the same star or other shared object independently.
I don’t see why you need to resolve any chirality in mathematics to have the concepts of right and left.
~Max
You don’t need to acknowledge the existence of an alien world to have the concept of right or left. Right and left are abstract concepts, the alien world is (presumably) concrete. It is the job of the alien, as a sentient being, to apply the abstract concept to their concrete reality. You can’t help them do that without also referring to the same concrete reality.
~Max
OP’s question was not “is it possible to have the concept of right or left”
OP’s question was “is it possible to explain the concept of right or left”
You don’t need to acknowledge the existance of an alien world to explain the concept of left and right, either.
~Max
Exactly. I actually tried to figure it out myself for about a week before asking the question here, btw.
In order to explain it without using specific objects or visual aids, the explanation must be universal. The alien world is just an analogy here to better illustrate the difficulty.
I assert that any abstract concept can be built entirely off of abstract references, but that no explanation applying an abstract concept to the universe can be made without reference to that universe.
~Max
But the alien’s world, even if it is so far away that we can only communicate “online”, still exists in the same physical space as ours. They could apply a concept of “left” and “right” to their world, but that concept may or may not be the same as ours.
I say, “here is a number line. It goes from 1 to 9. For any number x on the line, ‘left of’ means membership in the set of numbers less than x, and ‘right’ of means membership in the set of numbers right of x.”
This is what I have on my desk, which the alien cannot see:
1 2 3 4 5 6 7 8 9
This is what the alien has on its desk, which I cannot see:
6 5 2 1 9 7 8 4 3
Now I ask, is the number 3 left of the number 7?
The alien says, “yes”.
QED.
~Max
It could be even worse. For all we know, our hypothetical alien correspondents could do their writing boustrophedonically. Then what happens?
I transmit to the alien a black and white image, encoded as ones and zeroes. This is the image in my mind:
: ]
This is the image as ones and zeros on grid paper, on my desk:
0000000
0100110
0000010
0100110
0000000
This is the transmission:
00000000100110000001001001100000000
I tell the alien, “these signals should be interpreted as an ordered set of colored squares that make up a seven-by-seven tile mosaic. ‘Zero’ means a light square and ‘one’ means a dark square. Number each column, with column one being the column of the first square I transmitted, and column seven being the column of the seventh square. The eighth signal should correspond to a square adjacent to the first, ninth, and fifteenth.” &etc.
I say, “This picture represents the mouth and eyes of a human face, turned on its side. Do you see the column with two dark squares, which are not adjacent to any other dark squares? Those dark squares represent eyes.”
Alien says, “ok.”
I explain right and left. “With respect to this mosaic, ‘to the right of’ means membership in the set of all squares in lower numbered columns. And ‘to the left of’ means the set of all squares in higher numbered columns. The other five dark squares represent a human mouth. Are they to the right or left of the eyes?”.
The alien says, “Okay, I see the mouth to the right of the eyes.”
The alien sees this mosaic, which I cannot see:
07060504030201
07161504031201
07160504030201
07161504031201
07060504030201
Using the same system, I could transmit all sorts of pictures. We aren't actually looking at the same pictures because the alien applied abstract concepts differently than I did. Nevertheless, we can agree on how things are to the left or right of each other. I could send a picture of a person holding a paper in his right hand, and the alien would understand that the person is holding a paper in his right hand. We could go so far as to build a video game where the alien and I share virtual space, and when I say "snipers on the right!" the alien will respond appropriately.
~Max
This thread is beginning to resemble the old 0.999… = 1 “debate”.
And yet, if the two of you were to actually bump into each other on a distant planet, and before you even get a chance to say hello, the alien suddenly panics and says “Klathuri snipers behind you! Quick! Dodge right or else you’ll be molten slag!”, which way are you going to move?
At best, you create two systems that map to one another with useful consistency, but you haven’t communicated the real-world meaning of left and right.
If Czarcasm meant to ask whether one can explain the real-world meaning, and difference between, the directions “left” and “right” without referring to the real world, my answer would be no.
If the question is whether one can explain the concept of, and the difference between, the directions of “left” and “right” without referring to specific objects or visual aids, I think I have answered that question in the affirmative.
~Max